265:
16922:
44:
13398:
In the case of a simply connected bounded domain with smooth boundary, the
Riemann mapping function and all its derivatives extend by continuity to the closure of the domain. This can be proved using regularity properties of solutions of the Dirichlet boundary value problem, which follow either from
4082:
If a sequence of nowhere-vanishing holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is identically zero or the limit is nowhere-vanishing. If a sequence of univalent holomorphic functions on an open domain has a uniform limit on compacta, then either the
9128:
in 1950 and 1958. In a theorem on "boundary variation" (to distinguish it from "interior variation"), he derived a differential equation and inequality, that relied on a measure-theoretic characterisation of straight-line segments due to
Ughtred Shuttleworth Haslam-Jones from 1936. Haslam-Jones'
15468:
999:
The analog of the
Riemann mapping theorem for more complicated domains is not true. The next simplest case is of doubly connected domains (domains with a single hole). Any doubly connected domain except for the punctured disk and the punctured plane is conformally equivalent to some annulus
931:. His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see
5888:
13543:
This algorithm converges for Jordan regions in the sense of uniformly close boundaries. There are corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a
8495:
954:, was published in 1922 and it was rather shorter than the previous ones. In this proof, like in Riemann's proof, the desired mapping was obtained as the solution of an extremal problem. The Fejér–Riesz proof was further simplified by
863:
was able to prove that, to a large extent, the
Dirichlet principle is valid under the hypothesis that Riemann was working with. However, in order to be valid, the Dirichlet principle needs certain hypotheses concerning the boundary of
3009:
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8160:
8570:
As a consequence of the
Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk. If points are omitted, this follows from the theorem. For the whole plane, the homeomorphism
4180:
6407:
12048:
850:
wrote once, concerning the original formulation of the theorem, that it was “ultimately formulated in terms which would defy any attempt of proof, even with modern methods”. Riemann's flawed proof depended on the
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513:
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13956:
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1205:
The analogue of the
Riemann mapping theorem in three or more real dimensions is not true. The family of conformal maps in three dimensions is very poor, and essentially contains only
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6815:
589:
and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it.
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9555:
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Computational conformal mapping is prominently featured in problems of applied analysis and mathematical physics, as well as in engineering disciplines, such as image processing.
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gave a proof of uniformization for parallel slit domains which was similar to the
Riemann mapping theorem. To simplify notation, horizontal slits will be taken. Firstly, by
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7695:
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6308:
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4990:
7453:
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4202:. Hence winding numbers are continuous under uniform limits, so that if each function in the sequence has no zeros nor can the limit. For the second statement suppose that
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8656:
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391:
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4042:. But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in two less sides (with self-intersections permitted).
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1084:
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14015:
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12803:
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9854:
9691:
9475:
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1705:
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12662:
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11684:
11657:
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10337:
7500:
5468:
5441:
5109:
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3511:
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797:
777:
680:
640:
14687:
13400:
9053:
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15079:
14948:
6493:
2115:
3965:
3739:
15381:(1953), L. Ahlfors; E. Calabi; M. Morse; L. Sario; D. Spencer (eds.), "Developments of the Theory of Conformal Mapping and Riemann Surfaces Through a Century",
14925:
13888:
13384:
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7655:
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660:
613:
583:
563:
543:
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415:
334:
13404:
9136:
970:
Even relatively simple
Riemann mappings (for example a map from the interior of a circle to the interior of a square) have no explicit formula using only
2938:
11718:
8037:
3453:
with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from
295:
16448:
932:
11686:
and isolated points (the images of other the inverse of the
Joukowsky transform under the other parallel horizontal slits). Thus, taking a fixed
16564:
16882:
6322:
4093:
11917:
13711:
pixels, the oracle can say whether each pixel belongs to the boundary or not). Then A computes the absolute values of the uniformizing map
5883:{\displaystyle |g_{n}(z)-g_{m}(z)|\leq |g_{n}(z)-g_{n}(w_{i})|+|g_{n}(w_{i})-g_{m}(w_{i})|+|g_{m}(w_{1})-g_{m}(z)|\leq 4M\delta +2\delta .}
5182:
13581:. The algorithm was discovered as an approximate method for conformal welding; however, it can also be viewed as a discretization of the
10995:
1210:
11412:
16702:
4090:
If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number
16737:
15485:
Lakhtakia, Akhlesh; Varadan, Vijay K.; Messier, Russell (August 1987). "Generalisations and randomisation of the plane Koch curve".
9125:
11298:
5294:
16742:
4077:
288:
16948:
16302:
2390:
1239:
10894:
9129:
proof was regarded as difficult and was only given a satisfactory proof in the mid-1970s by
Schober and Campbell–Lamoureux.
927:
gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than
9862:
8856:
9787:
1845:
13468:
in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve
8969:
4069:
gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta.
16857:
16847:
16832:
16752:
15335:
11146:
453:
15876:
Marshall, Donald E.; Rohde, Steffen (2007). "Convergence of a Variant of the Zipper Algorithm for Conformal Mapping".
14789:
9264:
7875:
1199:
16892:
16532:
16434:
16382:
16357:
16275:
Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse
16187:
16166:
16145:
16124:
16099:
16020:
13491:
281:
146:
15913:
Binder, Ilia; Braverman, Mark; Yampolsky, Michael (2007). "On the computational complexity of the Riemann mapping".
14563:
13714:
13408:
7042:
16907:
10683:
9084:
The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by
8759:
6660:
1646:
15340:
10760:
8169:
15285:
15155:
13913:
12143:
11096:
9905:
4249:
16902:
16897:
16872:
16727:
16695:
16585:
16324:
16042:
16002:
15981:
15239:
12373:
11493:
4054:
The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives.
4010:
885:
16674:
16612:
16457:
14020:
13582:
13407:. Other methods for proving the smooth Riemann mapping theorem include the theory of kernel functions or the
12694:
10283:
6759:
3345:, such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point
16837:
16722:
14225:
13155:
9696:
9483:
9383:
9145:
8490:{\displaystyle F'(a)=H'(a)/(1-|H(a)|^{2})=f'(a)\left({\sqrt {|c|}}+{\sqrt {|c|^{-1}}}\right)/2>f'(a)=M.}
4066:
3516:
1327:
151:
141:
113:
14286:
12972:
11540:
10513:
10469:
9748:
2743:
16862:
16669:
16209:
15204:
13588:
The following is known about numerically approximating the conformal mapping between two planar domains.
13425:
7382:
3426:. The reasoning follows a "northeast argument": in the non self-intersecting path there will be a corner
1147:
1093:
1003:
10717:
9607:
8574:
7286:
6694:
5960:
2160:
16887:
16867:
16446:(1900), "On the Existence of the Green's Function for the Most General Simply Connected Plane Region",
14404:
13674:
7700:
1960:
14475:
14149:
14084:
13671:
is provided to A by an oracle representing it in a pixelated sense (i.e., if the screen is divided to
13651:
13618:
6820:
2858:
1730:
1245:
924:
16953:
16926:
16688:
16664:
14760:
14498:
13422:
In the early 1980s an elementary algorithm for computing conformal maps was discovered. Given points
12940:
10081:
8254:
7820:
7352:
6196:
205:
98:
89:
15890:
6416:
4995:
4329:
2692:
16877:
16767:
16732:
16505:
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse
14956:
14334:
10215:
9092:, on his book on univalent functions and conformal mappings, gave a treatment based on the work of
7137:
3594:
3456:
13344:
12312:
10869:
8639:
6315:
5473:
5114:
5061:
3255:
2582:
2516:
2470:
1206:
902:
374:
352:
6657:. It is a normal family by Montel's theorem. By the characterization of simple-connectivity, for
4658:
3148:
3081:
1057:
982:
127:
14107:
12567:
12379:
12243:
10574:
9996:
7660:
7600:
7236:
7212:
6620:
6501:
6273:
6109:
5576:
4959:
4893:
4632:
4608:
4544:
4489:
4465:
4437:
3904:
3218:
2775:
689:
16802:
15885:
13387:
10179:
9225:
9105:
9101:
7419:
7093:
5897:
4205:
2604:
1533:
340:
15084:
15013:
14731:
14527:
14440:
13961:
13837:
12895:
11878:
11232:
10833:
10019:
9347:
7186:
6970:
16827:
16231:
15974:
Complex analysis. An introduction to the theory of analytic functions of one complex variable
15120:
15036:
14879:
14859:
14692:
14384:
14199:
13893:
13598:
13471:
13084:
10382:
9793:
9121:
9058:
8949:
8793:
8713:
7794:
7269:
to produce a conformal mapping solving the problem: in this situation it is often called the
6895:
5556:
3568:
3198:
2280:
2240:
2047:
1897:
1782:
1274:
1087:
810:
which both lack at least two points of the sphere can be conformally mapped into each other.
729:
221:
196:
23:
16136:(1980), "Extremal problems for univalent functions", in Brannan, D.A.; Clunie, J.G. (eds.),
14626:
13990:
13807:
13777:
12773:
11689:
10055:
9833:
9670:
9454:
9113:
7565:
6944:
6585:
6238:
6074:
5988:
4486:
has a subsequence that converges to a holomorphic function uniformly on compacta. A family
1684:
16649:
16547:
Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse
16224:
15932:
15365:
15343:– a conformal transformation of the upper half-plane onto the interior of a simple polygon.
14172:
13547:
12848:
12640:
12508:
12481:
12285:
12110:
12083:
12056:
11662:
11635:
11268:
10636:
10402:
10315:
9117:
7478:
7318:
5446:
5419:
5087:
4806:
4779:
4728:
4701:
4517:
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3489:
3429:
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3348:
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3014:
2891:
2320:
2213:
2067:
896:
782:
755:
665:
618:
421:
231:
212:
166:
108:
16594:
Schober, Glenn (1975), "Appendix C. Schiffer's boundary variation and fundamental lemma",
16481:
16286:
14663:
9029:
8686:
Koebe's uniformization theorem for normal families also generalizes to yield uniformizers
6993:
6915:
6866:
8:
16852:
16762:
16747:
15056:
14930:
13131:
11590:
11409:
By the strict maximality for the slit mapping in the previous paragraph, we can see that
6472:
4666:
4062:
2094:
1232:
978:
971:
955:
892:
852:
819:
118:
80:
16352:, Cambridge Studies in Advanced Mathematics, vol. 107, Cambridge University Press,
16270:
15936:
14381:
There is an algorithm A′ that computes the uniformizing map in the following sense. Let
10446:. This applies in particular to the coefficient: so by compactness there is a univalent
9093:
4661:, it follows that the derivatives of a locally bounded family are also locally bounded.
3947:
3721:
16772:
16637:
16485:
16426:
16273:(1932), "Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche",
16247:
16072:
15990:
15948:
15922:
14910:
13873:
13595:
There is an algorithm A that computes the uniformizing map in the following sense. Let
13369:
13317:
13293:
13261:
13241:
13221:
13201:
13181:
13161:
13137:
13113:
13093:
13062:
13042:
12875:
12828:
12808:
12667:
12620:
12600:
12547:
12461:
12441:
12421:
12355:
12223:
11615:
11595:
11076:
10663:
10616:
10449:
10429:
10362:
10342:
10263:
10159:
10139:
9976:
9583:
9563:
9205:
8833:
8813:
8739:
8689:
8661:
8544:
8524:
8504:
8015:
7995:
7975:
7955:
7855:
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7752:
7732:
7640:
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7525:
7505:
7458:
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6739:
6565:
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6525:
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6156:
6054:
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6014:
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5536:
5516:
5399:
5160:
5140:
5041:
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4755:
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4408:
4388:
4185:
3990:
3970:
3927:
3884:
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3328:
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3108:
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2300:
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2193:
2140:
2120:
2024:
1937:
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1583:
1563:
1510:
1490:
1467:
1447:
1424:
1401:
1381:
1358:
1228:
867:
855:(which was named by Riemann himself), which was considered sound at the time. However,
825:
735:
645:
598:
568:
548:
528:
427:
418:
400:
319:
269:
176:
16411:
16200:, Translations of Mathematical Monographs, vol. 26, American Mathematical Society
16053:"Untersuchungen über die konformen Abbildungen von festen und veranderlichen Gebieten"
15498:
11581:
The proof of the uniqueness of the conformal parallel slit transformation is given in
16842:
16782:
16629:
16581:
16558:
16528:
16469:
16430:
16395:
16391:
16378:
16353:
16320:
16298:
16183:
16162:
16141:
16120:
16095:
16076:
16038:
16016:
15998:
15977:
13080:
3423:
977:
Simply connected open sets in the plane can be highly complicated, for instance, the
966:
The following points detail the uniqueness and power of the Riemann mapping theorem:
264:
191:
103:
75:
16496:
15976:, International Series in Pure and Applied Mathematics (3rd ed.), McGraw-Hill,
15952:
15010:
Consider the problem of computing the conformal radius of a simply-connected domain
9097:
4657:
if their functions are uniformly bounded on each compact disk. Differentiating the
592:
16777:
16757:
16711:
16621:
16541:
16512:
16477:
16461:
16407:
16366:
16345:
16282:
16257:
16175:
16064:
15940:
15895:
15494:
15346:
14951:
8810:
and bounded by finitely many Jordan contours, there is a unique univalent function
3004:{\displaystyle {\frac {1}{2\pi }}\int _{\partial C}\mathrm {d} \mathrm {arg} (z-a)}
947:
928:
856:
843:
309:
251:
246:
236:
65:
35:
11865:{\displaystyle F_{0}(w)=h\circ f(w)=(w-a)^{-1}+a_{1}(w-a)+a_{2}(w-a)^{2}+\cdots ,}
16822:
16797:
16645:
16607:
16599:
16577:
16524:
16338:
16316:
16220:
16161:, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag,
16116:
16091:
16034:
13311:
13287:
9109:
8155:{\displaystyle F(z)={\frac {e^{i\theta }(H(z)-H(a))}{1-{\overline {H(a)}}H(z)}},}
1311:
939:
171:
136:
16545:
15469:"What did Riemann Contribute to Mathematics? Geometry, Number Theory and Others"
11140:. On the other hand by the Riemann mapping theorem there is a conformal mapping
9667:
is characterized by an "extremal condition" as the unique univalent function in
16812:
16807:
16108:
16083:
13364:
13335:
6314:, the univalent holomorphic maps of the unit disk onto itself are given by the
4830:. It must be verified that this sequence of holomorphic functions converges on
1557:
988:
of infinite length, even if the set itself is bounded. One such example is the
807:
445:
226:
161:
156:
51:
16262:
15944:
16942:
16633:
16520:
16473:
13339:
13130:
and has the given boundary condition? The positive answer is provided by the
9601:
9085:
6311:
3644:
be the open rectangle with these vertices. The winding number of the path is
1217:
985:
951:
884:(namely, that it is a Jordan curve) which are not valid for simply connected
860:
800:
586:
347:
186:
181:
70:
16374:
15117:
pixels. Denote the problem of computing the conformal radius with precision
806:
As a corollary of the theorem, any two simply connected open subsets of the
16792:
16443:
16235:
15969:
15863:
15378:
12533:
7280:
1221:
847:
394:
12505:'s). This contradicts the fact that the non-constant holomorphic function
1301:
are both simply connected, but there is no biholomorphic map between them.
946:
two years later in a way that did not require them. Another proof, due to
16817:
16501:"Zum Parallelschlitztheorm unendlich- vielfach zusammenhängender Gebiete"
16418:
16205:
16154:
16133:
13578:
241:
60:
3924:
but not on the path, by continuity the winding number of the path about
16641:
16489:
16068:
10379:
is also in the family and each coefficient of the Laurent expansion at
7262:
6402:{\displaystyle k(z)=e^{i\theta }(z-\alpha )/(1-{\overline {\alpha }}z)}
4175:{\displaystyle {\frac {1}{2\pi i}}\int _{C}g^{-1}(z)g'(z)\mathrm {d} z}
989:
943:
15899:
12043:{\displaystyle F_{1}(w)=(w-a)^{-1}+b_{1}(w-a)+b_{2}(w-a)^{2}+\cdots .}
16398:(1987), "Generalisations and randomisation of the plane Koch curve",
15927:
14992:
14856:
Suppose there is an algorithm A that given a simply-connected domain
14560:
pixels. Then A′ computes the absolute values of the uniformizing map
5279:{\displaystyle g_{n}(b)-g_{n}(a)=\int _{a}^{b}g_{n}^{\prime }(z)\,dz}
337:
16625:
16465:
16052:
13258:
have been constructed, one has to check that the resulting function
12688:
is bounded and its boundary is smooth, much like Riemann did. Write
10633:
of its boundary which is not a horizontal slit. Then the complement
3325:
consecutive directed vertical and horizontal sides. By induction on
16680:
16252:
16217:
Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento
11066:{\displaystyle \mathrm {Re} (a_{1}+b_{1})\leq \mathrm {Re} (a_{1})}
9899:
can be uniformized by a horizontal parallel slit conformal mapping
3249:
2835:: it consists of finitely many horizontal and vertical segments in
1298:
1224:
343:
16337:, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 18,
15005:
is poly-time reducible to computing the conformal radius of a set.
11483:{\displaystyle \mathrm {Re} (c_{1})<\mathrm {Re} (b_{1}+c_{1})}
4671:
Every locally bounded family of holomorphic functions in a domain
15002:
13286:
The Riemann mapping theorem can be generalized to the context of
4725:
be a totally bounded sequence and chose a countable dense subset
2427:
can be written as the disjoint union of two open and closed sets
996:
manner to the nice and regular unit disc seems counter-intuitive.
16238:(2017), "The Riemann mapping theorem from Riemann's viewpoint",
15995:
Complex analysis.The argument principle in analysis and topology
13090:
The question then becomes: does a real-valued harmonic function
10312:, by Montel's theorem they form a normal family. Furthermore if
1090:
except inversion and multiplication by constants so the annulus
16297:, Studia Mathematica, vol. 4, Vandenhoeck & Ruprecht,
12805:
is some (to be determined) holomorphic function with real part
1681:(1) ⇒ (2) because any continuous closed curve, with base point
43:
16390:
10754:. By the Riemann mapping theorem there is a conformal mapping
7475:. By Montel's theorem, passing to a subsequence if necessary,
1779:(2) ⇒ (3) because the integral over any piecewise smooth path
16295:
Lectures on theory of functions in multiply connected domains
16013:
The Cauchy transform, potential theory, and conformal mapping
14728:)-machine). Furthermore, the algorithm computes the value of
3422:
by the induction hypothesis and elementary properties of the
2689:
be the compact set of the union of all squares with distance
14283:
then A can be used to compute the uniformizing map in space
12240:. If it is constant, then it must be identically zero since
11399:{\displaystyle f(k(w))-c_{0}=w+(a_{1}+c_{1})w^{-1}+\cdots .}
5389:{\displaystyle |g_{n}(a)-g_{n}(b)|\leq M|a-b|\leq 2\delta M}
859:
found that this principle was not universally valid. Later,
16369:(2006), "Riemann Mapping Theorem and its Generalizations",
14876:
with a linear-time computable boundary and an inner radius
12140:-coordinate so are horizontal segments. On the other hand,
6235:
would be a univalent holomorphic map of the unit disk with
3305:; such a rectangular path is determined by a succession of
15233:
14991:
then we can use one call to A to solve any instance of a
13059:
is the real part of a holomorphic function, we know that
4458:
of holomorphic functions on an open domain is said to be
3372:, then it breaks up into two rectangular paths of length
3252:
class as a rectangular path on the square grid of length
15912:
15484:
15364:
The existence of f is equivalent to the existence of a
9600:
so the denominator is nowhere-vanishing, and apply the
2420:{\displaystyle \mathbb {C} \cup \{\infty \}\setminus G}
682:
is an arbitrary angle, then there exists precisely one
15302:
14825:
14765:
14503:
11875:
with its image a horizontal slit domain. Suppose that
11612:
to the horizontal slit domain, it can be assumed that
10982:{\displaystyle h(f(z))=z+(a_{1}+b_{1})z^{-1}+\cdots ,}
10567:
is the required parallel slit transformation, suppose
3078:. On the other hand the sum of the winding numbers of
921:
itself; this established the Riemann mapping theorem.
818:
The theorem was stated (under the assumption that the
15288:
15242:
15207:
15158:
15123:
15087:
15059:
15039:
15016:
14959:
14933:
14913:
14882:
14862:
14792:
14763:
14734:
14695:
14666:
14629:
14566:
14530:
14501:
14478:
14443:
14407:
14387:
14337:
14289:
14228:
14202:
14175:
14152:
14110:
14087:
14023:
13993:
13964:
13916:
13896:
13876:
13840:
13810:
13780:
13717:
13677:
13654:
13621:
13601:
13550:
13494:
13474:
13428:
13372:
13347:
13320:
13296:
13264:
13244:
13224:
13204:
13184:
13164:
13140:
13116:
13096:
13065:
13045:
12975:
12943:
12898:
12878:
12851:
12831:
12811:
12776:
12697:
12670:
12643:
12623:
12603:
12570:
12550:
12539:
12511:
12484:
12464:
12444:
12424:
12382:
12358:
12315:
12288:
12246:
12226:
12146:
12113:
12086:
12059:
11920:
11881:
11721:
11692:
11665:
11638:
11618:
11598:
11543:
11496:
11415:
11301:
11271:
11235:
11149:
11099:
11079:
10998:
10897:
10872:
10836:
10763:
10720:
10686:
10666:
10639:
10619:
10577:
10516:
10472:
10452:
10432:
10405:
10385:
10365:
10345:
10318:
10286:
10266:
10218:
10182:
10162:
10142:
10084:
10058:
10022:
9999:
9979:
9908:
9892:{\displaystyle G\subset \mathbb {C} \cup \{\infty \}}
9865:
9836:
9796:
9751:
9699:
9673:
9610:
9586:
9566:
9486:
9457:
9386:
9350:
9267:
9228:
9208:
9148:
9061:
9032:
8972:
8952:
8936:{\displaystyle f(z)=z^{-1}+a_{1}z+a_{2}z^{2}+\cdots }
8859:
8836:
8816:
8796:
8762:
8742:
8716:
8692:
8664:
8642:
8577:
8547:
8527:
8507:
8289:
8257:
8172:
8040:
8018:
7998:
7978:
7958:
7878:
7858:
7823:
7797:
7777:
7755:
7735:
7703:
7663:
7643:
7603:
7568:
7548:
7528:
7508:
7481:
7461:
7422:
7385:
7355:
7321:
7289:
7239:
7215:
7189:
7140:
7096:
7076:
7045:
7025:
6996:
6973:
6947:
6918:
6898:
6869:
6823:
6762:
6742:
6697:
6663:
6623:
6588:
6568:
6548:
6528:
6504:
6475:
6455:
6419:
6325:
6276:
6241:
6199:
6179:
6159:
6112:
6077:
6057:
6037:
6017:
5991:
5963:
5933:
5900:
5614:
5579:
5559:
5539:
5519:
5476:
5449:
5422:
5402:
5297:
5185:
5163:
5143:
5117:
5090:
5064:
5044:
4998:
4962:
4942:
4922:
4896:
4876:
4856:
4836:
4809:
4782:
4758:
4731:
4704:
4677:
4635:
4611:
4591:
4571:
4547:
4520:
4492:
4468:
4440:
4411:
4391:
4332:
4252:
4208:
4188:
4096:
4013:
3993:
3973:
3950:
3930:
3907:
3887:
3867:
3847:
3827:
3807:
3787:
3767:
3747:
3724:
3697:
3670:
3664:
for points to the right of the vertical segment from
3650:
3630:
3597:
3571:
3519:
3492:
3459:
3432:
3401:
3378:
3351:
3331:
3311:
3284:
3258:
3221:
3201:
3195:(7) ⇒ (1) This is a purely topological argument. Let
3178:
3151:
3131:
3111:
3084:
3064:
3044:
3017:
2941:
2921:
2894:
2861:
2841:
2821:
2801:
2778:
2746:
2726:
2695:
2675:
2655:
2635:
2607:
2585:
2565:
2545:
2519:
2499:
2473:
2453:
2433:
2393:
2370:
2350:
2323:
2303:
2283:
2263:
2243:
2216:
2196:
2163:
2143:
2123:
2097:
2070:
2050:
2027:
1963:
1940:
1920:
1900:
1848:
1825:
1805:
1785:
1762:
1733:
1713:
1707:, can be continuously deformed to the constant curve
1687:
1649:
1629:
1606:
1586:
1566:
1536:
1513:
1493:
1470:
1450:
1427:
1404:
1384:
1361:
1330:
1277:
1248:
1150:
1096:
1060:
1006:
905:
891:
The first rigorous proof of the theorem was given by
870:
828:
785:
758:
738:
692:
668:
648:
621:
601:
571:
551:
531:
456:
430:
403:
377:
355:
322:
2855:
forming a finite number of closed rectangular paths
16198:
Geometric theory of functions of a complex variable
6522:to be the family of holomorphic univalent mappings
3395:, and thus can be deformed to the constant path at
1887:{\displaystyle f^{-1}\,\mathrm {d} f/\mathrm {d} z}
16610:(1973), "History of the Riemann mapping theorem",
15318:
15274:
15224:
15193:
15144:
15109:
15073:
15045:
15025:
14983:
14942:
14919:
14899:
14868:
14841:
14778:
14749:
14714:
14681:
14652:
14615:
14552:
14516:
14487:
14465:
14429:
14393:
14369:
14323:
14275:
14214:
14188:
14161:
14138:
14096:
14073:
14009:
13979:
13950:
13902:
13882:
13862:
13826:
13796:
13766:
13703:
13663:
13640:
13607:
13563:
13535:
13480:
13460:
13378:
13355:
13326:
13302:
13270:
13250:
13230:
13210:
13190:
13170:
13146:
13122:
13102:
13071:
13051:
13028:
12958:
12929:
12884:
12864:
12837:
12817:
12797:
12759:
12676:
12656:
12629:
12609:
12589:
12556:
12524:
12497:
12470:
12450:
12430:
12410:
12364:
12344:
12301:
12274:
12232:
12212:
12126:
12099:
12072:
12042:
11903:
11864:
11704:
11678:
11651:
11624:
11604:
11570:
11529:
11482:
11398:
11284:
11257:
11218:
11132:
11085:
11065:
10981:
10880:
10858:
10819:
10746:
10706:
10672:
10652:
10625:
10605:
10556:
10499:
10458:
10438:
10418:
10391:
10371:
10351:
10331:
10304:
10272:
10252:
10204:
10168:
10148:
10128:
10070:
10044:
10008:
9985:
9961:
9891:
9848:
9822:
9778:
9737:
9685:
9659:
9592:
9572:
9549:
9469:
9443:
9372:
9333:
9250:
9214:
9191:
9067:
9047:
9019:{\displaystyle \mathrm {Re} (e^{-2i\theta }a_{1})}
9018:
8958:
8935:
8842:
8822:
8802:
8782:
8748:
8722:
8698:
8670:
8650:
8628:
8553:
8533:
8513:
8489:
8273:
8243:
8154:
8024:
8004:
7984:
7964:
7944:
7864:
7844:
7809:
7783:
7761:
7741:
7721:
7689:
7649:
7629:
7589:
7554:
7534:
7514:
7494:
7467:
7447:
7408:
7371:
7341:
7307:
7249:
7225:
7201:
7175:
7126:
7082:
7062:
7031:
7011:
6982:
6959:
6933:
6904:
6884:
6855:
6809:
6748:
6728:
6683:
6649:
6609:
6574:
6554:
6534:
6514:
6487:
6461:
6441:
6401:
6302:
6262:
6227:
6185:
6165:
6138:
6098:
6063:
6043:
6023:
6003:
5977:
5939:
5919:
5882:
5598:
5565:
5545:
5525:
5505:
5462:
5435:
5408:
5388:
5278:
5169:
5149:
5129:
5103:
5076:
5050:
5030:
4984:
4948:
4928:
4908:
4882:
4862:
4842:
4822:
4795:
4764:
4744:
4717:
4683:
4645:
4621:
4597:
4577:
4557:
4533:
4502:
4478:
4450:
4417:
4397:
4377:
4318:
4238:
4194:
4174:
4034:
3999:
3979:
3959:
3936:
3916:
3893:
3873:
3853:
3833:
3813:
3793:
3773:
3753:
3733:
3710:
3683:
3656:
3636:
3616:
3583:
3557:
3505:
3478:
3445:
3414:
3387:
3364:
3337:
3317:
3297:
3270:
3240:
3207:
3184:
3164:
3145:. Hence the winding number of at least one of the
3137:
3117:
3097:
3070:
3050:
3030:
3003:
2927:
2907:
2880:
2847:
2827:
2807:
2787:
2764:
2732:
2712:
2681:
2661:
2641:
2621:
2593:
2571:
2551:
2531:
2505:
2485:
2459:
2439:
2419:
2376:
2356:
2336:
2309:
2289:
2269:
2249:
2229:
2202:
2182:
2149:
2129:
2109:
2083:
2056:
2033:
2013:
1946:
1926:
1906:
1886:
1831:
1811:
1791:
1768:
1748:
1719:
1699:
1669:
1635:
1612:
1592:
1572:
1548:
1519:
1499:
1476:
1456:
1433:
1410:
1390:
1367:
1344:
1289:
1263:
1190:
1136:
1078:
1046:
913:
876:
834:
791:
771:
744:
720:
674:
654:
634:
607:
577:
557:
537:
507:
436:
409:
385:
363:
328:
16449:Transactions of the American Mathematical Society
13958:Furthermore, the algorithm computes the value of
13393:
13310:is a non-empty simply-connected open subset of a
11219:{\displaystyle k(w)=w+c_{0}+c_{1}w^{-1}+\cdots ,}
7729:. It remains to check that the conformal mapping
6691:there is a holomorphic branch of the square root
899:on arbitrary simply connected domains other than
508:{\displaystyle D=\{z\in \mathbb {C} :|z|<1\}.}
16940:
16315:, Springer-Lehrbuch (in German) (3rd ed.),
16182:, Undergraduate Texts in Mathematics, Springer,
16033:, Graduate Texts in Mathematics, vol. 125,
15419:had already proven the Riemann mapping theorem).
14842:{\displaystyle |\phi (w)|<1-{\tfrac {1}{n}}.}
14301:
10888:with a horizontal slit removed. So we have that
9859:To prove now that the multiply connected domain
9334:{\displaystyle f(z)=z+a_{0}+a_{1}z^{-1}+\cdots }
7945:{\displaystyle (f(z)-c)/(1-{\overline {c}}f(z))}
5084:is taken small enough, finitely many open disks
424:mapping whose inverse is also holomorphic) from
16598:, Lecture Notes in Mathematics, vol. 478,
16210:"On the history of the Riemann mapping theorem"
15415:, p. 108, footnote ** (acknowledging that
15383:Contributions to the Theory of Riemann Surfaces
15001:) with a linear time overhead. In other words,
13536:{\displaystyle z_{0},\ldots ,z_{n}\in \gamma .}
9790:, applied to the family of univalent functions
5927:forms a Cauchy sequence in the uniform norm on
1238:The analogue of the Riemann mapping theorem in
1086:, however there are no conformal maps between
992:. The fact that such a set can be mapped in an
16400:Journal of Physics A: Mathematical and General
16015:, Studies in Advanced Mathematics, CRC Press,
15487:Journal of Physics A: Mathematical and General
14616:{\displaystyle \phi :(\Omega ,w_{0})\to (D,0)}
13767:{\displaystyle \phi :(\Omega ,w_{0})\to (D,0)}
13334:is biholomorphic to one of the following: the
13278:does indeed have all the required properties.
13198:(this argument depends on the assumption that
9100:from the early 1930s; it was the precursor of
7063:{\displaystyle \mathbb {C} \setminus -\Delta }
1227:can be found that are not homeomorphic to the
16696:
16028:
15875:
15566:
7522:uniformly on compacta. By Hurwitz's theorem,
1487:every nowhere-vanishing holomorphic function
1444:every nowhere-vanishing holomorphic function
1144:is not conformally equivalent to the annulus
615:is unique up to rotation and recentering: if
289:
16738:Grothendieck–Hirzebruch–Riemann–Roch theorem
16050:
15869:
15432:
15412:
12458:will eventually be encircled by contours in
10707:{\displaystyle \mathbb {C} \cup \{\infty \}}
10701:
10695:
10299:
10293:
9886:
9880:
8783:{\displaystyle \mathbb {C} \cup \{\infty \}}
8777:
8771:
6684:{\displaystyle b\in \mathbb {C} \setminus G}
5914:
5901:
4979:
4963:
4326:. These are nowhere-vanishing on a disk but
4083:limit is constant or the limit is univalent.
3215:be a piecewise smooth closed curve based at
2408:
2402:
1670:{\displaystyle \mathbb {C} \cup \{\infty \}}
1664:
1658:
1441:is the derivative of a holomorphic function;
1305:
1185:
1151:
1131:
1097:
1041:
1007:
565:does not contain any “holes”. The fact that
499:
463:
16517:Classical topics in complex function theory
10820:{\displaystyle h(w)=w+b_{1}w^{-1}+\cdots ,}
8244:{\displaystyle H'(a)/|H'(a)|=e^{-i\theta }}
7039:be the unique Möbius transformation taking
2539:be the shortest Euclidean distance between
2387:(6) ⇒ (7) for otherwise the extended plane
1378:the integral of every holomorphic function
16703:
16689:
16563:: CS1 maint: location missing publisher (
16335:Univalent functions and conformal mapping.
16029:Berenstein, Carlos A.; Gay, Roger (1991),
15319:{\displaystyle 0<c<{\tfrac {1}{2}}.}
15194:{\displaystyle {\texttt {CONF}}(n,n^{c}).}
14401:be a bounded simply-connected domain, and
13951:{\displaystyle d(w_{0},\partial \Omega ).}
13615:be a bounded simply-connected domain, and
12213:{\displaystyle F_{2}(w)=F_{0}(w)-F_{1}(w)}
11133:{\displaystyle \mathrm {Re} (b_{1})\leq 0}
10426:tends to the corresponding coefficient of
9962:{\displaystyle f(z)=z+a_{1}z^{-1}+\cdots }
4319:{\displaystyle g_{n}(z)=f_{n}(z)-f_{n}(a)}
4045:
3741:for points to the right; and hence inside
1398:around a closed piecewise smooth curve in
296:
282:
16883:Riemann–Roch theorem for smooth manifolds
16662:
16495:
16261:
16251:
16230:
15926:
15889:
15443:
15408:
15275:{\displaystyle {\texttt {CONF}}(n,n^{c})}
13349:
13281:
11530:{\displaystyle \mathrm {Re} (b_{1})>0}
10874:
10688:
9873:
9124:, summarised in addresses he gave to the
8764:
8706:for multiply-connected domains to finite
8681:
8644:
7047:
6671:
5971:
5269:
4035:{\displaystyle R\cup \partial R\subset G}
3011:equals the sum of the winding numbers of
2587:
2395:
1862:
1737:
1651:
1580:for any piecewise smooth closed curve in
1352:the following conditions are equivalent:
1338:
1251:
907:
473:
379:
357:
16269:
16138:Aspects of contemporary complex analysis
15906:
15245:
15211:
15161:
14169:is polynomial space computable in space
9126:International Congress of Mathematicians
16593:
16540:
16511:
16394:; Varadan, Vijay K.; Messier, Russell;
16344:
16332:
16292:
16195:
16174:
15989:
15968:
15839:
15827:
15803:
15780:
15736:
15724:
15689:
15678:
15644:
15589:
15578:
15554:
15541:
15531:
15526:
15511:
15377:
14074:{\displaystyle |\phi (w)|<1-2^{-n}.}
12760:{\displaystyle f(z)=(z-z_{0})e^{g(z)},}
12684:. For this sketch, we will assume that
11632:is a domain bounded by the unit circle
11586:
11582:
9120:gave a treatment based on very general
9089:
4776:", a subsequence can be chosen so that
3861:is a point of the path, it must lie in
1957:(4) ⇒ (5) by taking the square root as
16941:
16571:
16442:
16417:
16365:
16310:
16107:
16082:
15815:
15792:
15769:
15712:
15667:
15633:
15611:
15536:
15416:
15396:
12376:shows that the number of solutions of
10613:has a compact and connected component
10305:{\displaystyle G\setminus \{\infty \}}
9786:: this is an immediate consequence of
9132:
9108:, later developed as the technique of
6810:{\displaystyle h(z_{1})\neq -h(z_{2})}
6011:, there is a unique conformal mapping
2041:is a holomorphic choice of logarithm.
1316:
585:is biholomorphic implies that it is a
16684:
16606:
16153:
16132:
15862:A Jordan region is the interior of a
15758:
15747:
15701:
15656:
15622:
15600:
15400:
14276:{\displaystyle T(n)<2^{O(n^{a})},}
9738:{\displaystyle z+a_{1}z^{-1}+\cdots }
9550:{\displaystyle g(z)=S(f(S/z)-b)^{-1}}
9444:{\displaystyle |f(z)-a_{0}|\leq 2|z|}
9192:{\displaystyle g(z)=z+cz^{2}+\cdots }
8281:and a routine computation shows that
7542:is either univalent or constant. But
3558:{\displaystyle w_{0}=z_{0}-in\delta }
1345:{\displaystyle G\subset \mathbb {C} }
799:. This is an easy consequence of the
16710:
16204:
16140:, Academic Press, pp. 181–208,
16113:Functions of one complex variable II
16010:
15851:
15557:, pp. 256–257, elementary proof
15455:
15428:
15404:
14324:{\displaystyle C\cdot n^{\max(a,2)}}
13029:{\displaystyle u(z)=-\log |z-z_{0}|}
12309:is non-constant, then by assumption
11571:{\displaystyle \mathrm {Re} (b_{1})}
10557:{\displaystyle f(z)=z+a_{1}+\cdots }
10500:{\displaystyle \mathrm {Re} (a_{1})}
9779:{\displaystyle \mathrm {Re} (a_{1})}
4061:This is an immediate consequence of
3248:. By approximation γ is in the same
2765:{\displaystyle C\cap B=\varnothing }
1220:in higher dimensions are permitted,
895:in 1900. He proved the existence of
16596:Univalent functions—selected topics
15478:
15399:. For accounts of the history, see
15225:{\displaystyle {\texttt {MAJ}}_{n}}
13461:{\displaystyle z_{0},\ldots ,z_{n}}
9222:in the open unit disk must satisfy
8501:This contradicts the maximality of
7409:{\displaystyle f_{n}\in {\cal {F}}}
1954:to give a branch of the logarithm.
1839:can be used to define a primitive.
1191:{\displaystyle \{z:1<|z|<4\}}
1137:{\displaystyle \{z:1<|z|<2\}}
1047:{\displaystyle \{z:r<|z|<1\}}
13:
16848:Riemannian connection on a surface
16753:Measurable Riemann mapping theorem
16313:Funktionentheorie. Eine Einführung
16240:Complex Analysis and Its Synergies
16031:Complex variables. An introduction
15878:SIAM Journal on Numerical Analysis
15336:Measurable Riemann mapping theorem
15040:
15017:
14966:
14863:
14576:
14482:
14479:
14421:
14388:
14156:
14153:
14091:
14088:
13939:
13936:
13897:
13727:
13658:
13655:
13635:
13602:
12950:
12597:, we want to construct a function
12540:Sketch proof via Dirichlet problem
11548:
11545:
11501:
11498:
11447:
11444:
11420:
11417:
11104:
11101:
11043:
11040:
11003:
11000:
10747:{\displaystyle G_{2}\supset G_{1}}
10698:
10477:
10474:
10386:
10296:
10000:
9883:
9756:
9753:
9660:{\displaystyle f_{R}(z)=z+R^{2}/z}
9055:a parallel slit domain with angle
8977:
8974:
8953:
8797:
8774:
8629:{\displaystyle \phi (x)=z/(1+|z|)}
8266:
7713:
7401:
7364:
7308:{\displaystyle 0<M\leq \infty }
7302:
7242:
7218:
7057:
6977:
6899:
6729:{\displaystyle h(z)={\sqrt {z-b}}}
6507:
5978:{\displaystyle G\neq \mathbb {C} }
5255:
4638:
4614:
4550:
4495:
4471:
4443:
4165:
4020:
3908:
2982:
2979:
2976:
2971:
2962:
2779:
2474:
2405:
2183:{\displaystyle f_{n}\circ \gamma }
1877:
1864:
1739:
1661:
14:
16965:
16656:
16613:The American Mathematical Monthly
16088:Functions of one complex variable
14430:{\displaystyle w_{0}\in \Omega .}
13704:{\displaystyle 2^{n}\times 2^{n}}
13401:Sobolev spaces for planar domains
10290:
7827:
7722:{\displaystyle {f\in {\cal {F}}}}
7051:
6675:
5985:is a simply connected domain and
4052:Weierstrass' convergence theorem.
2759:
2411:
2014:{\displaystyle g(z)=\exp(f(x)/2)}
16921:
16920:
14488:{\displaystyle \partial \Omega }
14162:{\displaystyle \partial \Omega }
14097:{\displaystyle \partial \Omega }
13890:depends only on the diameter of
13664:{\displaystyle \partial \Omega }
13641:{\displaystyle w_{0}\in \Omega }
13110:exist that is defined on all of
10260:. Since the family of univalent
7872:be a holomorphic square root of
7502:tends to a holomorphic function
6856:{\displaystyle z_{1},z_{2}\in G}
4772:. By locally boundedness and a "
4462:if any sequence of functions in
2881:{\displaystyle \gamma _{j}\in G}
2064:is a piecewise closed curve and
1749:{\displaystyle f\,\mathrm {d} z}
1264:{\displaystyle \mathbb {C} ^{n}}
525:Intuitively, the condition that
263:
42:
16833:Riemann's differential equation
16743:Hirzebruch–Riemann–Roch theorem
15856:
15845:
15833:
15821:
15809:
15797:
15786:
15774:
15763:
15752:
15741:
15730:
15718:
15706:
15695:
15683:
15672:
15661:
15650:
15638:
15627:
15616:
15605:
15594:
15583:
15572:
15560:
15548:
15517:
14779:{\displaystyle {\tfrac {1}{n}}}
14660:in randomized space bounded by
14517:{\displaystyle {\tfrac {1}{n}}}
12959:{\displaystyle z\in \partial U}
11712:, there is a univalent mapping
10129:{\displaystyle |f(z)|\leq 2|z|}
8274:{\displaystyle F\in {\cal {F}}}
7845:{\displaystyle D\setminus f(G)}
7372:{\displaystyle f\in {\cal {F}}}
6228:{\displaystyle h=f\circ g^{-1}}
6193:satisfied the same conditions,
2915:to be all the squares covering
2669:at the centre of a square. Let
2091:are successive square roots of
545:be simply connected means that
16858:Riemann–Hilbert correspondence
16728:Generalized Riemann hypothesis
15505:
15461:
15449:
15437:
15422:
15389:
15371:
15358:
15269:
15250:
15185:
15166:
15104:
15091:
14975:
14963:
14811:
14807:
14801:
14794:
14744:
14738:
14676:
14670:
14647:
14633:
14610:
14598:
14595:
14592:
14573:
14547:
14534:
14495:is given to A′ with precision
14359:
14346:
14316:
14304:
14265:
14252:
14238:
14232:
14128:
14122:
14042:
14038:
14032:
14025:
13974:
13968:
13942:
13920:
13855:
13849:
13761:
13749:
13746:
13743:
13724:
13394:Smooth Riemann mapping theorem
13022:
13001:
12985:
12979:
12917:
12913:
12907:
12900:
12749:
12743:
12732:
12713:
12707:
12701:
12399:
12393:
12372:is not in one of these lines,
12339:
12326:
12263:
12257:
12207:
12201:
12185:
12179:
12163:
12157:
12022:
12009:
11993:
11981:
11956:
11943:
11937:
11931:
11898:
11892:
11844:
11831:
11815:
11803:
11778:
11765:
11759:
11753:
11738:
11732:
11589:. Applying the inverse of the
11565:
11552:
11518:
11505:
11477:
11451:
11437:
11424:
11371:
11345:
11320:
11317:
11311:
11305:
11245:
11237:
11159:
11153:
11121:
11108:
11060:
11047:
11033:
11007:
10954:
10928:
10916:
10913:
10907:
10901:
10853:
10840:
10773:
10767:
10587:
10581:
10526:
10520:
10494:
10481:
10339:is in the family and tends to
10237:
10233:
10227:
10220:
10192:
10184:
10122:
10114:
10103:
10099:
10093:
10086:
10078:, univalency and the estimate
10032:
10024:
9918:
9912:
9809:
9800:
9773:
9760:
9627:
9621:
9535:
9525:
9511:
9505:
9496:
9490:
9437:
9429:
9418:
9401:
9395:
9388:
9360:
9352:
9277:
9271:
9238:
9230:
9158:
9152:
9042:
9036:
9013:
8981:
8869:
8863:
8623:
8619:
8611:
8601:
8587:
8581:
8475:
8469:
8430:
8421:
8409:
8401:
8390:
8384:
8370:
8360:
8355:
8349:
8342:
8332:
8324:
8318:
8304:
8298:
8218:
8214:
8208:
8196:
8187:
8181:
8143:
8137:
8125:
8119:
8102:
8099:
8093:
8084:
8078:
8072:
8050:
8044:
7939:
7936:
7930:
7908:
7900:
7891:
7885:
7879:
7839:
7833:
7678:
7672:
7618:
7612:
7578:
7572:
7442:
7436:
7336:
7330:
7164:
7161:
7155:
7149:
7115:
7112:
7106:
7100:
7006:
7000:
6928:
6922:
6879:
6873:
6804:
6791:
6779:
6766:
6707:
6701:
6638:
6632:
6598:
6592:
6442:{\displaystyle |\alpha |<1}
6429:
6421:
6396:
6374:
6366:
6354:
6335:
6329:
6291:
6285:
6251:
6245:
6153:Uniqueness follows because if
6127:
6121:
6087:
6081:
5852:
5848:
5842:
5826:
5813:
5799:
5791:
5787:
5774:
5758:
5745:
5731:
5723:
5719:
5706:
5690:
5684:
5670:
5662:
5658:
5652:
5636:
5630:
5616:
5500:
5487:
5370:
5356:
5345:
5341:
5335:
5319:
5313:
5299:
5266:
5260:
5224:
5218:
5202:
5196:
5031:{\displaystyle |g_{n}'|\leq M}
5018:
5000:
4378:{\displaystyle g(z)=f(z)-f(a)}
4372:
4366:
4357:
4351:
4342:
4336:
4313:
4307:
4291:
4285:
4269:
4263:
4233:
4227:
4218:
4212:
4161:
4155:
4144:
4138:
3761:. Since the winding number is
2998:
2986:
2713:{\displaystyle \leq \delta /4}
2277:. Hence the winding number of
2008:
1997:
1991:
1985:
1973:
1967:
1643:in the extended complex plane
1527:has a holomorphic square root;
1421:every holomorphic function in
1175:
1167:
1121:
1113:
1031:
1023:
709:
696:
489:
481:
1:
16893:Riemann–Siegel theta function
16458:American Mathematical Society
15962:
15081:by an explicit collection of
14984:{\displaystyle r(\Omega ,0),}
14370:{\displaystyle 2^{O(n^{a})}.}
13583:Loewner differential equation
13414:
13158:for the holomorphic function
12352:are all horizontal lines. If
10253:{\displaystyle |f(z)|\leq 2S}
8710:, where the slits have angle
7176:{\displaystyle F'(h(a))>0}
6469:must be the identity map and
3617:{\displaystyle w_{0}-\delta }
3479:{\displaystyle z_{0}-\delta }
2157:, then the winding number of
961:
16949:Theorems in complex analysis
16908:Riemann–von Mangoldt formula
15395:For the original paper, see
13356:{\displaystyle \mathbb {C} }
12345:{\displaystyle F_{2}(C_{i})}
11911:is another uniformizer with
10881:{\displaystyle \mathbb {C} }
10359:uniformly on compacta, then
8651:{\displaystyle \mathbb {C} }
8129:
7922:
6388:
5506:{\displaystyle g_{n}(w_{i})}
5130:{\displaystyle \delta >0}
5077:{\displaystyle \delta >0}
4850:uniformly on each compactum
4803:is convergent at each point
3271:{\displaystyle \delta >0}
2594:{\displaystyle \mathbb {C} }
2532:{\displaystyle \delta >0}
2486:{\displaystyle \infty \in B}
2237:times the winding number of
1484:has a holomorphic logarithm;
1200:proven using extremal length
914:{\displaystyle \mathbb {C} }
846:in 1851 in his PhD thesis.
386:{\displaystyle \mathbb {C} }
364:{\displaystyle \mathbb {C} }
7:
16670:Encyclopedia of Mathematics
16412:10.1088/0305-4470/20/11/052
15499:10.1088/0305-4470/20/11/052
15341:Schwarz–Christoffel mapping
15329:
13218:be simply connected). Once
12374:Cauchy's argument principle
11659:and contains analytic arcs
11537:. The two inequalities for
9580:in the unit disk, choosing
6892:must contain a closed disk
4565:and converges uniformly to
4182:for a holomorphic function
3591:and then goes leftwards to
3165:{\displaystyle \gamma _{j}}
3098:{\displaystyle \gamma _{j}}
2579:and build a square grid on
1727:. So the line integral of
1079:{\displaystyle 0<r<1}
518:This mapping is known as a
10:
16970:
16903:Riemann–Stieltjes integral
16898:Riemann–Silberstein vector
16873:Riemann–Liouville integral
16333:Jenkins, James A. (1958),
14139:{\displaystyle 2^{-O(n)}.}
14104:with precision of at most
13405:classical potential theory
13154:has been established, the
12590:{\displaystyle z_{0}\in U}
12411:{\displaystyle F_{2}(w)=t}
12275:{\displaystyle F_{2}(a)=0}
10606:{\displaystyle f(G)=G_{1}}
10009:{\displaystyle \partial G}
7690:{\displaystyle f'(a)>0}
7630:{\displaystyle f'(a)>0}
7250:{\displaystyle {\cal {F}}}
7226:{\displaystyle {\cal {F}}}
6650:{\displaystyle f'(a)>0}
6515:{\displaystyle {\cal {F}}}
6303:{\displaystyle h'(0)>0}
6139:{\displaystyle f'(a)>0}
5599:{\displaystyle z\in D_{k}}
4985:{\displaystyle \{g_{n}'\}}
4909:{\displaystyle K\subset E}
4646:{\displaystyle {\cal {F}}}
4622:{\displaystyle {\cal {F}}}
4558:{\displaystyle {\cal {F}}}
4503:{\displaystyle {\cal {F}}}
4479:{\displaystyle {\cal {F}}}
4451:{\displaystyle {\cal {F}}}
3917:{\displaystyle \partial R}
3241:{\displaystyle z_{0}\in G}
2788:{\displaystyle \partial C}
1309:
938:Carathéodory's proof used
813:
721:{\displaystyle f(z_{0})=0}
16916:
16838:Riemann's minimal surface
16718:
16663:Dolzhenko, E.P. (2001) ,
16371:Geometric Function Theory
16263:10.1186/s40627-016-0009-7
16051:Carathéodory, C. (1912),
15997:, John Wiley & Sons,
15945:10.1007/s11512-007-0045-x
15567:Berenstein & Gay 1991
15431:, pp. 78–80, citing
10714:is simply connected with
10205:{\displaystyle |z|\leq S}
9251:{\displaystyle |c|\leq 2}
9139:, any univalent function
8636:gives a homeomorphism of
7448:{\displaystyle f_{n}'(a)}
7127:{\displaystyle F(h(a))=0}
6498:To prove existence, take
5920:{\displaystyle \{g_{n}\}}
4916:such that the closure of
4239:{\displaystyle f(a)=f(b)}
4067:Cauchy's integral formula
4065:for the first statement.
2622:{\displaystyle \delta /4}
1842:(3) ⇒ (4) by integrating
1549:{\displaystyle w\notin G}
1306:Proof via normal families
1240:several complex variables
942:and it was simplified by
206:Geometric function theory
152:Cauchy's integral formula
142:Cauchy's integral theorem
16863:Riemann–Hilbert problems
16768:Riemann curvature tensor
16733:Grand Riemann hypothesis
16723:Cauchy–Riemann equations
16572:Schiff, Joel L. (1993),
16293:Grunsky, Helmut (1978),
16011:Bell, Steven R. (1992),
15352:
15110:{\displaystyle O(n^{2})}
15053:is given with precision
15026:{\displaystyle \Omega ,}
14750:{\displaystyle \phi (w)}
14553:{\displaystyle O(n^{2})}
14466:{\displaystyle n=2^{k},}
13980:{\displaystyle \phi (w)}
13863:{\displaystyle 2^{O(n)}}
13156:Cauchy–Riemann equations
13134:. Once the existence of
12930:{\displaystyle |f(z)|=1}
12845:. It is then clear that
11904:{\displaystyle F_{1}(w)}
11258:{\displaystyle |w|>S}
10859:{\displaystyle h(G_{2})}
10045:{\displaystyle |z|<R}
9373:{\displaystyle |z|>R}
8541:must take all values in
7202:{\displaystyle F\circ h}
6983:{\displaystyle -\Delta }
6562:into the open unit disk
5955:Riemann mapping theorem.
4936:is compact and contains
4425:must vanish identically.
842:is piecewise smooth) by
114:Cauchy–Riemann equations
16788:Riemann mapping theorem
16456:(3), Providence, R.I.:
16196:Goluzin, G. M. (1969),
15145:{\displaystyle 1/n^{c}}
15046:{\displaystyle \Omega }
14900:{\displaystyle >1/2}
14869:{\displaystyle \Omega }
14715:{\displaystyle n=2^{k}}
14689:and time polynomial in
14394:{\displaystyle \Omega }
14215:{\displaystyle a\geq 1}
13903:{\displaystyle \Omega }
13608:{\displaystyle \Omega }
13481:{\displaystyle \gamma }
13386:. This is known as the
13039:on the boundary. Since
10392:{\displaystyle \infty }
10280:are locally bounded in
9823:{\displaystyle f(zR)/R}
9788:Grönwall's area theorem
9258:. As a consequence, if
9137:Bieberbach's inequality
9106:quadratic differentials
9102:quasiconformal mappings
9068:{\displaystyle \theta }
8959:{\displaystyle \infty }
8803:{\displaystyle \infty }
8723:{\displaystyle \theta }
7810:{\displaystyle c\neq 0}
7090:with the normalization
6905:{\displaystyle \Delta }
5573:of its limit. Then for
5566:{\displaystyle \delta }
4659:Cauchy integral formula
4514:if whenever a sequence
4046:Riemann mapping theorem
3584:{\displaystyle n\geq 1}
3208:{\displaystyle \gamma }
2290:{\displaystyle \gamma }
2250:{\displaystyle \gamma }
2057:{\displaystyle \gamma }
1907:{\displaystyle \gamma }
1792:{\displaystyle \gamma }
1290:{\displaystyle n\geq 2}
925:Constantin Carathéodory
314:Riemann mapping theorem
99:Complex-valued function
16888:Riemann–Siegel formula
16868:Riemann–Lebesgue lemma
16803:Riemann series theorem
16553:(in German), Göttingen
16311:Jänich, Klaus (1993),
15411:, p. 4. Also see
15320:
15276:
15226:
15195:
15146:
15111:
15075:
15047:
15033:where the boundary of
15027:
14985:
14944:
14921:
14901:
14870:
14843:
14780:
14751:
14716:
14683:
14654:
14653:{\displaystyle O(1/n)}
14617:
14554:
14518:
14489:
14467:
14437:Suppose that for some
14431:
14395:
14371:
14325:
14277:
14216:
14190:
14163:
14140:
14098:
14075:
14011:
14010:{\displaystyle 2^{-n}}
13981:
13952:
13904:
13884:
13864:
13828:
13827:{\displaystyle Cn^{2}}
13798:
13797:{\displaystyle 2^{-n}}
13768:
13705:
13665:
13642:
13609:
13565:
13537:
13482:
13462:
13388:uniformization theorem
13380:
13357:
13328:
13304:
13282:Uniformization theorem
13272:
13252:
13232:
13212:
13192:
13172:
13148:
13124:
13104:
13073:
13053:
13030:
12960:
12931:
12886:
12866:
12839:
12819:
12799:
12798:{\displaystyle g=u+iv}
12761:
12678:
12658:
12631:
12611:
12591:
12558:
12526:
12499:
12472:
12452:
12432:
12412:
12366:
12346:
12303:
12276:
12234:
12214:
12128:
12101:
12074:
12044:
11905:
11866:
11706:
11705:{\displaystyle a\in G}
11680:
11653:
11626:
11606:
11572:
11531:
11484:
11400:
11286:
11259:
11220:
11134:
11087:
11073:by the extremality of
11067:
10983:
10882:
10860:
10821:
10748:
10708:
10674:
10654:
10627:
10607:
10558:
10501:
10460:
10440:
10420:
10393:
10373:
10353:
10333:
10306:
10274:
10254:
10206:
10170:
10150:
10130:
10072:
10071:{\displaystyle S>R}
10046:
10016:lies in the open disk
10010:
9987:
9963:
9893:
9850:
9849:{\displaystyle z>1}
9824:
9780:
9739:
9687:
9686:{\displaystyle z>R}
9661:
9594:
9574:
9551:
9471:
9470:{\displaystyle S>R}
9445:
9374:
9335:
9252:
9216:
9193:
9122:variational principles
9069:
9049:
9020:
8960:
8937:
8844:
8824:
8804:
8784:
8750:
8724:
8700:
8682:Parallel slit mappings
8672:
8652:
8630:
8555:
8535:
8515:
8491:
8275:
8245:
8156:
8026:
8006:
7992:is univalent and maps
7986:
7966:
7946:
7866:
7846:
7811:
7785:
7763:
7743:
7723:
7691:
7651:
7631:
7591:
7590:{\displaystyle f(a)=0}
7556:
7536:
7516:
7496:
7469:
7449:
7410:
7373:
7343:
7309:
7251:
7227:
7203:
7177:
7128:
7084:
7064:
7033:
7013:
6984:
6961:
6960:{\displaystyle r>0}
6935:
6906:
6886:
6857:
6811:
6756:. It is univalent and
6750:
6730:
6685:
6651:
6611:
6610:{\displaystyle f(a)=0}
6576:
6556:
6536:
6516:
6489:
6463:
6443:
6403:
6316:Möbius transformations
6304:
6264:
6263:{\displaystyle h(0)=0}
6229:
6187:
6167:
6140:
6100:
6099:{\displaystyle f(a)=0}
6065:
6045:
6025:
6005:
6004:{\displaystyle a\in G}
5979:
5941:
5921:
5884:
5600:
5567:
5553:so large to be within
5547:
5527:
5507:
5464:
5437:
5410:
5390:
5280:
5171:
5151:
5137:are required to cover
5131:
5105:
5078:
5052:
5032:
4986:
4950:
4930:
4910:
4884:
4864:
4844:
4824:
4797:
4766:
4746:
4719:
4685:
4647:
4623:
4599:
4579:
4559:
4535:
4504:
4480:
4452:
4419:
4399:
4379:
4320:
4240:
4196:
4176:
4036:
4001:
3981:
3961:
3938:
3918:
3895:
3875:
3855:
3835:
3815:
3795:
3775:
3755:
3735:
3712:
3685:
3658:
3638:
3618:
3585:
3559:
3507:
3480:
3447:
3416:
3389:
3366:
3339:
3319:
3299:
3272:
3242:
3209:
3186:
3166:
3139:
3119:
3099:
3072:
3052:
3032:
3005:
2929:
2909:
2882:
2849:
2829:
2809:
2789:
2766:
2734:
2714:
2683:
2663:
2643:
2623:
2595:
2573:
2553:
2533:
2507:
2487:
2461:
2441:
2421:
2378:
2358:
2338:
2311:
2291:
2271:
2251:
2231:
2204:
2184:
2151:
2131:
2111:
2085:
2058:
2035:
2015:
1948:
1928:
1908:
1888:
1833:
1813:
1793:
1770:
1750:
1721:
1701:
1700:{\displaystyle a\in G}
1671:
1637:
1614:
1594:
1574:
1550:
1521:
1501:
1478:
1458:
1435:
1412:
1392:
1369:
1346:
1291:
1265:
1207:Möbius transformations
1192:
1138:
1080:
1048:
933:Carathéodory's theorem
915:
878:
836:
793:
773:
746:
722:
676:
656:
636:
609:
579:
559:
539:
509:
438:
411:
393:, then there exists a
387:
365:
330:
270:Mathematics portal
16828:Riemann zeta function
16057:Mathematische Annalen
15514:, section 8.3, p. 187
15444:Greene & Kim 2017
15409:Greene & Kim 2017
15321:
15277:
15227:
15196:
15147:
15112:
15076:
15048:
15028:
14986:
14945:
14922:
14902:
14871:
14844:
14781:
14752:
14717:
14684:
14655:
14618:
14555:
14519:
14490:
14468:
14432:
14396:
14372:
14326:
14278:
14217:
14191:
14189:{\displaystyle n^{a}}
14164:
14141:
14099:
14076:
14012:
13982:
13953:
13905:
13885:
13865:
13829:
13799:
13769:
13706:
13666:
13643:
13610:
13566:
13564:{\displaystyle C^{1}}
13538:
13483:
13463:
13381:
13358:
13329:
13305:
13273:
13253:
13233:
13213:
13193:
13173:
13149:
13125:
13105:
13083:; i.e., it satisfies
13074:
13054:
13031:
12961:
12932:
12887:
12867:
12865:{\displaystyle z_{0}}
12840:
12820:
12800:
12762:
12679:
12659:
12657:{\displaystyle z_{0}}
12637:to the unit disk and
12632:
12612:
12592:
12559:
12527:
12525:{\displaystyle F_{2}}
12500:
12498:{\displaystyle C_{i}}
12473:
12453:
12433:
12413:
12367:
12347:
12304:
12302:{\displaystyle F_{2}}
12277:
12235:
12215:
12129:
12127:{\displaystyle C_{i}}
12102:
12100:{\displaystyle F_{1}}
12075:
12073:{\displaystyle F_{0}}
12045:
11906:
11867:
11707:
11681:
11679:{\displaystyle C_{i}}
11654:
11652:{\displaystyle C_{0}}
11627:
11607:
11573:
11532:
11485:
11401:
11287:
11285:{\displaystyle G_{2}}
11260:
11221:
11135:
11088:
11068:
10984:
10883:
10861:
10822:
10749:
10709:
10675:
10655:
10653:{\displaystyle G_{2}}
10628:
10608:
10559:
10502:
10461:
10441:
10421:
10419:{\displaystyle f_{n}}
10394:
10374:
10354:
10334:
10332:{\displaystyle f_{n}}
10307:
10275:
10255:
10207:
10171:
10151:
10131:
10073:
10047:
10011:
9988:
9964:
9894:
9851:
9825:
9781:
9740:
9688:
9662:
9595:
9575:
9552:
9472:
9446:
9375:
9336:
9253:
9217:
9194:
9070:
9050:
9021:
8961:
8938:
8845:
8825:
8805:
8785:
8751:
8725:
8708:parallel slit domains
8701:
8673:
8653:
8631:
8556:
8536:
8516:
8492:
8276:
8246:
8157:
8027:
8007:
7987:
7967:
7947:
7867:
7847:
7812:
7786:
7764:
7744:
7724:
7692:
7652:
7632:
7592:
7557:
7537:
7517:
7497:
7495:{\displaystyle f_{n}}
7470:
7450:
7411:
7374:
7344:
7342:{\displaystyle f'(a)}
7310:
7252:
7228:
7204:
7178:
7129:
7085:
7065:
7034:
7014:
6985:
6962:
6936:
6907:
6887:
6858:
6812:
6751:
6731:
6686:
6652:
6612:
6577:
6557:
6537:
6517:
6490:
6464:
6444:
6404:
6305:
6265:
6230:
6188:
6168:
6141:
6101:
6071:normalized such that
6066:
6046:
6026:
6006:
5980:
5942:
5922:
5885:
5601:
5568:
5548:
5528:
5508:
5465:
5463:{\displaystyle D_{k}}
5438:
5436:{\displaystyle w_{i}}
5411:
5391:
5281:
5172:
5152:
5132:
5106:
5104:{\displaystyle D_{k}}
5079:
5058:. By compactness, if
5053:
5033:
4987:
4956:. Since the sequence
4951:
4931:
4911:
4885:
4865:
4845:
4825:
4823:{\displaystyle w_{m}}
4798:
4796:{\displaystyle g_{n}}
4767:
4747:
4745:{\displaystyle w_{m}}
4720:
4718:{\displaystyle f_{n}}
4686:
4648:
4624:
4600:
4580:
4560:
4536:
4534:{\displaystyle f_{n}}
4505:
4481:
4453:
4420:
4400:
4380:
4321:
4241:
4197:
4177:
4037:
4002:
3982:
3962:
3939:
3919:
3896:
3876:
3856:
3836:
3816:
3796:
3776:
3756:
3736:
3713:
3711:{\displaystyle w_{0}}
3686:
3684:{\displaystyle z_{0}}
3659:
3639:
3619:
3586:
3560:
3508:
3506:{\displaystyle z_{0}}
3481:
3448:
3446:{\displaystyle z_{0}}
3417:
3415:{\displaystyle z_{1}}
3390:
3388:{\displaystyle <N}
3367:
3365:{\displaystyle z_{1}}
3340:
3320:
3300:
3298:{\displaystyle z_{0}}
3273:
3243:
3210:
3187:
3167:
3140:
3120:
3100:
3073:
3053:
3033:
3031:{\displaystyle C_{i}}
3006:
2930:
2910:
2908:{\displaystyle C_{i}}
2883:
2850:
2830:
2810:
2790:
2767:
2735:
2715:
2684:
2664:
2644:
2624:
2596:
2574:
2554:
2534:
2508:
2488:
2462:
2442:
2422:
2379:
2359:
2339:
2337:{\displaystyle 2^{n}}
2317:must be divisible by
2312:
2292:
2272:
2252:
2232:
2230:{\displaystyle 2^{n}}
2205:
2185:
2152:
2132:
2112:
2086:
2084:{\displaystyle f_{n}}
2059:
2044:(5) ⇒ (6) because if
2036:
2016:
1949:
1929:
1909:
1889:
1834:
1814:
1794:
1771:
1751:
1722:
1702:
1672:
1638:
1615:
1595:
1575:
1551:
1522:
1502:
1479:
1459:
1436:
1413:
1393:
1370:
1347:
1292:
1266:
1242:is also not true. In
1193:
1139:
1081:
1049:
958:and by Carathéodory.
916:
879:
837:
794:
792:{\displaystyle \phi }
774:
772:{\displaystyle z_{0}}
747:
732:of the derivative of
723:
677:
675:{\displaystyle \phi }
657:
637:
635:{\displaystyle z_{0}}
610:
580:
560:
540:
510:
439:
412:
388:
366:
331:
222:Augustin-Louis Cauchy
24:Mathematical analysis
16878:Riemann–Roch theorem
16507:(in German): 199−202
16176:Gamelin, Theodore W.
15403:, pp. 270–271;
15286:
15240:
15205:
15156:
15121:
15085:
15057:
15037:
15014:
14957:
14931:
14911:
14880:
14860:
14790:
14761:
14732:
14693:
14682:{\displaystyle O(k)}
14664:
14627:
14564:
14528:
14499:
14476:
14441:
14405:
14385:
14335:
14287:
14226:
14200:
14173:
14150:
14108:
14085:
14081:Moreover, A queries
14021:
13991:
13962:
13914:
13894:
13874:
13838:
13808:
13804:in space bounded by
13778:
13715:
13675:
13652:
13619:
13599:
13548:
13492:
13472:
13426:
13370:
13345:
13318:
13294:
13262:
13242:
13222:
13202:
13182:
13162:
13138:
13114:
13094:
13063:
13043:
12973:
12941:
12896:
12876:
12872:is the only zero of
12849:
12829:
12809:
12774:
12695:
12668:
12641:
12621:
12601:
12568:
12548:
12509:
12482:
12462:
12442:
12422:
12380:
12356:
12313:
12286:
12244:
12224:
12144:
12111:
12084:
12057:
11918:
11879:
11719:
11690:
11663:
11636:
11616:
11596:
11541:
11494:
11413:
11299:
11269:
11233:
11147:
11097:
11077:
10996:
10895:
10870:
10834:
10761:
10718:
10684:
10664:
10637:
10617:
10575:
10569:reductio ad absurdum
10514:
10470:
10450:
10430:
10403:
10383:
10363:
10343:
10316:
10284:
10264:
10216:
10180:
10160:
10140:
10082:
10056:
10020:
9997:
9977:
9906:
9863:
9834:
9794:
9749:
9697:
9671:
9608:
9604:. Next the function
9584:
9564:
9484:
9455:
9451:. To see this, take
9384:
9348:
9265:
9226:
9206:
9146:
9059:
9048:{\displaystyle f(G)}
9030:
8970:
8950:
8857:
8834:
8814:
8794:
8760:
8740:
8714:
8690:
8662:
8640:
8575:
8545:
8525:
8505:
8287:
8255:
8170:
8038:
8016:
7996:
7976:
7956:
7876:
7856:
7821:
7795:
7775:
7753:
7733:
7701:
7661:
7657:is finite, equal to
7641:
7601:
7566:
7546:
7526:
7506:
7479:
7459:
7420:
7383:
7353:
7319:
7287:
7237:
7213:
7187:
7138:
7094:
7074:
7043:
7023:
7012:{\displaystyle h(G)}
6994:
6971:
6945:
6934:{\displaystyle h(a)}
6916:
6896:
6885:{\displaystyle h(G)}
6867:
6821:
6760:
6740:
6695:
6661:
6621:
6586:
6566:
6546:
6526:
6502:
6473:
6453:
6417:
6323:
6274:
6239:
6197:
6177:
6157:
6110:
6075:
6055:
6035:
6015:
5989:
5961:
5931:
5898:
5612:
5577:
5557:
5537:
5517:
5474:
5447:
5420:
5400:
5295:
5183:
5161:
5141:
5115:
5088:
5062:
5042:
4996:
4992:is locally bounded,
4960:
4940:
4920:
4894:
4874:
4854:
4834:
4807:
4780:
4756:
4729:
4702:
4675:
4633:
4609:
4589:
4569:
4545:
4518:
4490:
4466:
4438:
4409:
4389:
4330:
4250:
4206:
4186:
4094:
4011:
3991:
3971:
3948:
3928:
3905:
3885:
3865:
3845:
3825:
3805:
3785:
3765:
3745:
3722:
3695:
3668:
3648:
3628:
3595:
3569:
3517:
3490:
3457:
3430:
3399:
3376:
3349:
3329:
3309:
3282:
3256:
3219:
3199:
3176:
3149:
3129:
3109:
3082:
3062:
3042:
3015:
2939:
2919:
2892:
2859:
2839:
2819:
2799:
2776:
2744:
2724:
2693:
2673:
2653:
2633:
2605:
2583:
2563:
2543:
2517:
2497:
2471:
2451:
2431:
2391:
2368:
2348:
2321:
2301:
2281:
2261:
2241:
2214:
2194:
2161:
2141:
2121:
2095:
2068:
2048:
2025:
1961:
1938:
1918:
1898:
1846:
1823:
1803:
1783:
1760:
1731:
1711:
1685:
1647:
1627:
1604:
1584:
1564:
1534:
1511:
1491:
1468:
1448:
1425:
1402:
1382:
1375:is simply connected;
1359:
1328:
1275:
1246:
1148:
1094:
1058:
1004:
972:elementary functions
903:
868:
826:
783:
756:
736:
690:
666:
646:
619:
599:
595:proved that the map
569:
549:
529:
454:
428:
401:
375:
371:which is not all of
353:
348:complex number plane
320:
232:Carl Friedrich Gauss
167:Isolated singularity
109:Holomorphic function
16:Mathematical theorem
16853:Riemannian geometry
16763:Riemann Xi function
16748:Local zeta function
16377:, pp. 83–108,
16159:Univalent functions
15937:2007ArM....45..221B
15915:Arkiv för Matematik
15074:{\displaystyle 1/n}
14943:{\displaystyle 20n}
14927:computes the first
14722:(that is, by a BPL(
14623:within an error of
13132:Dirichlet principle
12825:and imaginary part
11591:Joukowsky transform
11578:are contradictory.
7435:
7315:be the supremum of
6488:{\displaystyle f=g}
6051:onto the unit disk
5894:Hence the sequence
5259:
5244:
5157:while remaining in
5016:
4978:
2364:, so it must equal
2110:{\displaystyle z-w}
1324:For an open domain
1317:Simple connectivity
1233:Whitehead continuum
1211:Liouville's theorem
956:Alexander Ostrowski
893:William Fogg Osgood
853:Dirichlet principle
686:as above such that
119:Formal power series
81:Unit complex number
16773:Riemann hypothesis
16602:, pp. 181–190
16427:Dover Publications
16396:Varadan, Vasundara
16392:Lakhtakia, Akhlesh
16069:10.1007/bf01456892
15842:, pp. 390–407
15830:, pp. 214−215
15818:, pp. 351–358
15806:, pp. 210–216
15783:, pp. 210–216
15715:, pp. 162–166
15407:, pp. 64–65;
15316:
15311:
15272:
15222:
15191:
15142:
15107:
15071:
15043:
15023:
14981:
14940:
14917:
14897:
14866:
14852:Negative results:
14839:
14834:
14776:
14774:
14747:
14712:
14679:
14650:
14613:
14550:
14514:
14512:
14485:
14463:
14427:
14391:
14367:
14321:
14273:
14212:
14196:for some constant
14186:
14159:
14146:In particular, if
14136:
14094:
14071:
14007:
13977:
13948:
13900:
13880:
13860:
13824:
13794:
13764:
13701:
13661:
13638:
13605:
13591:Positive results:
13561:
13533:
13478:
13458:
13376:
13353:
13324:
13300:
13268:
13248:
13228:
13208:
13188:
13168:
13144:
13120:
13100:
13085:Laplace's equation
13069:
13049:
13026:
12956:
12927:
12882:
12862:
12835:
12815:
12795:
12757:
12674:
12654:
12627:
12607:
12587:
12554:
12522:
12495:
12468:
12448:
12428:
12408:
12362:
12342:
12299:
12272:
12230:
12220:is holomorphic in
12210:
12124:
12097:
12070:
12040:
11901:
11862:
11702:
11676:
11649:
11622:
11602:
11568:
11527:
11480:
11396:
11282:
11255:
11216:
11130:
11083:
11063:
10979:
10878:
10856:
10817:
10744:
10704:
10670:
10650:
10623:
10603:
10554:
10497:
10456:
10436:
10416:
10389:
10369:
10349:
10329:
10302:
10270:
10250:
10202:
10166:
10146:
10126:
10068:
10042:
10006:
9993:large enough that
9983:
9959:
9889:
9846:
9820:
9776:
9735:
9683:
9657:
9590:
9570:
9547:
9467:
9441:
9370:
9331:
9248:
9212:
9189:
9114:Oswald Teichmüller
9065:
9045:
9016:
8956:
8933:
8840:
8820:
8800:
8780:
8746:
8720:
8696:
8668:
8648:
8626:
8551:
8531:
8511:
8487:
8271:
8241:
8152:
8022:
8002:
7982:
7962:
7942:
7862:
7842:
7807:
7781:
7759:
7739:
7719:
7687:
7647:
7627:
7587:
7552:
7532:
7512:
7492:
7465:
7445:
7423:
7406:
7369:
7339:
7305:
7247:
7223:
7199:
7183:. By construction
7173:
7124:
7080:
7060:
7029:
7009:
6980:
6957:
6931:
6902:
6882:
6853:
6807:
6746:
6726:
6681:
6647:
6607:
6572:
6552:
6532:
6512:
6485:
6459:
6439:
6399:
6300:
6260:
6225:
6183:
6163:
6136:
6096:
6061:
6041:
6021:
6001:
5975:
5937:
5917:
5880:
5596:
5563:
5543:
5523:
5503:
5460:
5433:
5406:
5386:
5276:
5245:
5230:
5167:
5147:
5127:
5101:
5074:
5048:
5028:
5004:
4982:
4966:
4946:
4926:
4906:
4880:
4860:
4840:
4820:
4793:
4762:
4742:
4715:
4681:
4643:
4619:
4595:
4585:on compacta, then
4575:
4555:
4531:
4500:
4476:
4448:
4415:
4395:
4375:
4316:
4236:
4192:
4172:
4032:
3997:
3977:
3960:{\displaystyle -1}
3957:
3934:
3914:
3891:
3871:
3851:
3831:
3811:
3791:
3771:
3751:
3734:{\displaystyle -1}
3731:
3708:
3681:
3654:
3634:
3614:
3581:
3555:
3503:
3476:
3443:
3412:
3385:
3362:
3335:
3315:
3295:
3268:
3238:
3205:
3182:
3162:
3135:
3115:
3095:
3068:
3048:
3028:
3001:
2925:
2905:
2878:
2845:
2825:
2805:
2785:
2762:
2730:
2710:
2679:
2659:
2639:
2619:
2591:
2569:
2549:
2529:
2503:
2483:
2457:
2437:
2417:
2374:
2354:
2334:
2307:
2287:
2267:
2247:
2227:
2200:
2180:
2147:
2127:
2107:
2081:
2054:
2031:
2011:
1944:
1924:
1904:
1884:
1829:
1809:
1789:
1766:
1756:over the curve is
1746:
1717:
1697:
1667:
1633:
1623:the complement of
1610:
1590:
1570:
1546:
1517:
1497:
1474:
1454:
1431:
1408:
1388:
1365:
1342:
1287:
1261:
1216:Even if arbitrary
1188:
1134:
1076:
1044:
911:
874:
832:
789:
769:
742:
728:and such that the
718:
672:
652:
632:
605:
575:
555:
535:
505:
434:
407:
383:
361:
326:
197:Laplace's equation
177:Argument principle
16936:
16935:
16843:Riemannian circle
16783:Riemann invariant
16665:"Riemann theorem"
16542:Riemann, Bernhard
16513:Remmert, Reinhold
16423:Conformal mapping
16406:(11): 3537–3541,
16367:Krantz, Steven G.
16346:Kodaira, Kunihiko
16304:978-3-525-40142-2
16271:Grötzsch, Herbert
16232:Greene, Robert E.
15900:10.1137/060659119
15493:(11): 3537–3541.
15433:Carathéodory 1912
15413:Carathéodory 1912
15310:
15247:
15213:
15163:
14920:{\displaystyle n}
14833:
14773:
14511:
13883:{\displaystyle C}
13409:Beltrami equation
13379:{\displaystyle D}
13327:{\displaystyle U}
13303:{\displaystyle U}
13271:{\displaystyle f}
13251:{\displaystyle v}
13231:{\displaystyle u}
13211:{\displaystyle U}
13191:{\displaystyle v}
13178:allow us to find
13171:{\displaystyle g}
13147:{\displaystyle u}
13123:{\displaystyle U}
13103:{\displaystyle u}
13081:harmonic function
13079:is necessarily a
13072:{\displaystyle u}
13052:{\displaystyle u}
12885:{\displaystyle f}
12838:{\displaystyle v}
12818:{\displaystyle u}
12677:{\displaystyle 0}
12630:{\displaystyle U}
12610:{\displaystyle f}
12557:{\displaystyle U}
12471:{\displaystyle G}
12451:{\displaystyle t}
12431:{\displaystyle G}
12365:{\displaystyle t}
12233:{\displaystyle G}
12053:The images under
11625:{\displaystyle G}
11605:{\displaystyle h}
11086:{\displaystyle f}
10673:{\displaystyle K}
10626:{\displaystyle K}
10459:{\displaystyle f}
10439:{\displaystyle f}
10372:{\displaystyle f}
10352:{\displaystyle f}
10273:{\displaystyle f}
10169:{\displaystyle G}
10149:{\displaystyle z}
9986:{\displaystyle R}
9593:{\displaystyle b}
9573:{\displaystyle z}
9215:{\displaystyle z}
9026:and having image
8843:{\displaystyle G}
8823:{\displaystyle f}
8749:{\displaystyle G}
8699:{\displaystyle f}
8671:{\displaystyle D}
8554:{\displaystyle D}
8534:{\displaystyle f}
8514:{\displaystyle M}
8443:
8413:
8147:
8132:
8025:{\displaystyle D}
8005:{\displaystyle G}
7985:{\displaystyle H}
7965:{\displaystyle G}
7925:
7865:{\displaystyle H}
7784:{\displaystyle D}
7762:{\displaystyle G}
7742:{\displaystyle f}
7650:{\displaystyle M}
7555:{\displaystyle f}
7535:{\displaystyle f}
7515:{\displaystyle f}
7468:{\displaystyle M}
7267:extremal function
7083:{\displaystyle D}
7032:{\displaystyle F}
6749:{\displaystyle G}
6724:
6575:{\displaystyle D}
6555:{\displaystyle G}
6535:{\displaystyle f}
6462:{\displaystyle h}
6391:
6186:{\displaystyle g}
6166:{\displaystyle f}
6064:{\displaystyle D}
6044:{\displaystyle G}
6024:{\displaystyle f}
5940:{\displaystyle K}
5546:{\displaystyle m}
5526:{\displaystyle n}
5409:{\displaystyle k}
5170:{\displaystyle E}
5150:{\displaystyle K}
5051:{\displaystyle E}
4949:{\displaystyle G}
4929:{\displaystyle E}
4883:{\displaystyle E}
4863:{\displaystyle K}
4843:{\displaystyle G}
4774:diagonal argument
4765:{\displaystyle G}
4684:{\displaystyle G}
4598:{\displaystyle f}
4578:{\displaystyle f}
4418:{\displaystyle g}
4398:{\displaystyle b}
4195:{\displaystyle g}
4113:
4078:Hurwitz's theorem
4000:{\displaystyle G}
3987:must also lie in
3980:{\displaystyle z}
3937:{\displaystyle z}
3894:{\displaystyle z}
3874:{\displaystyle G}
3854:{\displaystyle z}
3834:{\displaystyle G}
3814:{\displaystyle R}
3794:{\displaystyle G}
3774:{\displaystyle 0}
3754:{\displaystyle R}
3657:{\displaystyle 0}
3637:{\displaystyle R}
3424:fundamental group
3338:{\displaystyle N}
3318:{\displaystyle N}
3185:{\displaystyle a}
3138:{\displaystyle 1}
3118:{\displaystyle a}
3071:{\displaystyle 1}
3051:{\displaystyle a}
2955:
2928:{\displaystyle A}
2848:{\displaystyle G}
2828:{\displaystyle B}
2808:{\displaystyle A}
2733:{\displaystyle A}
2682:{\displaystyle C}
2662:{\displaystyle A}
2642:{\displaystyle a}
2572:{\displaystyle B}
2552:{\displaystyle A}
2506:{\displaystyle A}
2460:{\displaystyle B}
2440:{\displaystyle A}
2377:{\displaystyle 0}
2357:{\displaystyle n}
2310:{\displaystyle w}
2270:{\displaystyle 0}
2203:{\displaystyle w}
2150:{\displaystyle G}
2130:{\displaystyle w}
2034:{\displaystyle f}
1947:{\displaystyle x}
1927:{\displaystyle a}
1832:{\displaystyle z}
1812:{\displaystyle a}
1769:{\displaystyle 0}
1720:{\displaystyle a}
1636:{\displaystyle G}
1613:{\displaystyle 0}
1593:{\displaystyle G}
1573:{\displaystyle w}
1520:{\displaystyle G}
1500:{\displaystyle g}
1477:{\displaystyle G}
1457:{\displaystyle f}
1434:{\displaystyle G}
1411:{\displaystyle G}
1391:{\displaystyle f}
1368:{\displaystyle G}
981:can be a nowhere-
877:{\displaystyle U}
835:{\displaystyle U}
745:{\displaystyle f}
655:{\displaystyle U}
642:is an element of
608:{\displaystyle f}
578:{\displaystyle f}
558:{\displaystyle U}
538:{\displaystyle U}
437:{\displaystyle U}
410:{\displaystyle f}
329:{\displaystyle U}
306:
305:
192:Harmonic function
104:Analytic function
90:Complex functions
76:Complex conjugate
16961:
16954:Bernhard Riemann
16924:
16923:
16778:Riemann integral
16758:Riemann (crater)
16712:Bernhard Riemann
16705:
16698:
16691:
16682:
16681:
16677:
16652:
16603:
16590:
16576:, Universitext,
16568:
16562:
16554:
16552:
16537:
16519:, translated by
16508:
16492:
16439:
16414:
16387:
16362:
16350:Complex analysis
16341:
16329:
16307:
16289:
16266:
16265:
16255:
16227:
16214:
16201:
16192:
16180:Complex analysis
16171:
16150:
16129:
16104:
16079:
16047:
16025:
16007:
15991:Beardon, Alan F.
15986:
15970:Ahlfors, Lars V.
15957:
15956:
15930:
15910:
15904:
15903:
15893:
15873:
15867:
15860:
15854:
15849:
15843:
15837:
15831:
15825:
15819:
15813:
15807:
15801:
15795:
15790:
15784:
15778:
15772:
15767:
15761:
15756:
15750:
15745:
15739:
15734:
15728:
15727:, pp. 77–78
15722:
15716:
15710:
15704:
15699:
15693:
15692:, pp. 77–78
15687:
15681:
15676:
15670:
15665:
15659:
15654:
15648:
15642:
15636:
15631:
15625:
15620:
15614:
15609:
15603:
15598:
15592:
15587:
15581:
15576:
15570:
15569:, pp. 86–87
15564:
15558:
15552:
15546:
15521:
15515:
15509:
15503:
15502:
15482:
15476:
15475:
15473:
15465:
15459:
15458:, pp. 80–83
15453:
15447:
15441:
15435:
15426:
15420:
15393:
15387:
15386:
15375:
15369:
15366:Green’s function
15362:
15347:Conformal radius
15325:
15323:
15322:
15317:
15312:
15303:
15281:
15279:
15278:
15273:
15268:
15267:
15249:
15248:
15231:
15229:
15228:
15223:
15221:
15220:
15215:
15214:
15200:
15198:
15197:
15192:
15184:
15183:
15165:
15164:
15151:
15149:
15148:
15143:
15141:
15140:
15131:
15116:
15114:
15113:
15108:
15103:
15102:
15080:
15078:
15077:
15072:
15067:
15052:
15050:
15049:
15044:
15032:
15030:
15029:
15024:
15000:
14990:
14988:
14987:
14982:
14952:conformal radius
14949:
14947:
14946:
14941:
14926:
14924:
14923:
14918:
14906:
14904:
14903:
14898:
14893:
14875:
14873:
14872:
14867:
14848:
14846:
14845:
14840:
14835:
14826:
14814:
14797:
14785:
14783:
14782:
14777:
14775:
14766:
14756:
14754:
14753:
14748:
14727:
14721:
14719:
14718:
14713:
14711:
14710:
14688:
14686:
14685:
14680:
14659:
14657:
14656:
14651:
14643:
14622:
14620:
14619:
14614:
14591:
14590:
14559:
14557:
14556:
14551:
14546:
14545:
14523:
14521:
14520:
14515:
14513:
14504:
14494:
14492:
14491:
14486:
14472:
14470:
14469:
14464:
14459:
14458:
14436:
14434:
14433:
14428:
14417:
14416:
14400:
14398:
14397:
14392:
14376:
14374:
14373:
14368:
14363:
14362:
14358:
14357:
14330:
14328:
14327:
14322:
14320:
14319:
14282:
14280:
14279:
14274:
14269:
14268:
14264:
14263:
14221:
14219:
14218:
14213:
14195:
14193:
14192:
14187:
14185:
14184:
14168:
14166:
14165:
14160:
14145:
14143:
14142:
14137:
14132:
14131:
14103:
14101:
14100:
14095:
14080:
14078:
14077:
14072:
14067:
14066:
14045:
14028:
14016:
14014:
14013:
14008:
14006:
14005:
13986:
13984:
13983:
13978:
13957:
13955:
13954:
13949:
13932:
13931:
13909:
13907:
13906:
13901:
13889:
13887:
13886:
13881:
13869:
13867:
13866:
13861:
13859:
13858:
13833:
13831:
13830:
13825:
13823:
13822:
13803:
13801:
13800:
13795:
13793:
13792:
13773:
13771:
13770:
13765:
13742:
13741:
13710:
13708:
13707:
13702:
13700:
13699:
13687:
13686:
13670:
13668:
13667:
13662:
13647:
13645:
13644:
13639:
13631:
13630:
13614:
13612:
13611:
13606:
13576:
13570:
13568:
13567:
13562:
13560:
13559:
13542:
13540:
13539:
13534:
13523:
13522:
13504:
13503:
13487:
13485:
13484:
13479:
13467:
13465:
13464:
13459:
13457:
13456:
13438:
13437:
13385:
13383:
13382:
13377:
13362:
13360:
13359:
13354:
13352:
13333:
13331:
13330:
13325:
13309:
13307:
13306:
13301:
13288:Riemann surfaces
13277:
13275:
13274:
13269:
13257:
13255:
13254:
13249:
13237:
13235:
13234:
13229:
13217:
13215:
13214:
13209:
13197:
13195:
13194:
13189:
13177:
13175:
13174:
13169:
13153:
13151:
13150:
13145:
13129:
13127:
13126:
13121:
13109:
13107:
13106:
13101:
13078:
13076:
13075:
13070:
13058:
13056:
13055:
13050:
13035:
13033:
13032:
13027:
13025:
13020:
13019:
13004:
12965:
12963:
12962:
12957:
12936:
12934:
12933:
12928:
12920:
12903:
12891:
12889:
12888:
12883:
12871:
12869:
12868:
12863:
12861:
12860:
12844:
12842:
12841:
12836:
12824:
12822:
12821:
12816:
12804:
12802:
12801:
12796:
12766:
12764:
12763:
12758:
12753:
12752:
12731:
12730:
12683:
12681:
12680:
12675:
12663:
12661:
12660:
12655:
12653:
12652:
12636:
12634:
12633:
12628:
12616:
12614:
12613:
12608:
12596:
12594:
12593:
12588:
12580:
12579:
12563:
12561:
12560:
12555:
12531:
12529:
12528:
12523:
12521:
12520:
12504:
12502:
12501:
12496:
12494:
12493:
12477:
12475:
12474:
12469:
12457:
12455:
12454:
12449:
12437:
12435:
12434:
12429:
12417:
12415:
12414:
12409:
12392:
12391:
12371:
12369:
12368:
12363:
12351:
12349:
12348:
12343:
12338:
12337:
12325:
12324:
12308:
12306:
12305:
12300:
12298:
12297:
12281:
12279:
12278:
12273:
12256:
12255:
12239:
12237:
12236:
12231:
12219:
12217:
12216:
12211:
12200:
12199:
12178:
12177:
12156:
12155:
12139:
12133:
12131:
12130:
12125:
12123:
12122:
12106:
12104:
12103:
12098:
12096:
12095:
12079:
12077:
12076:
12071:
12069:
12068:
12049:
12047:
12046:
12041:
12030:
12029:
12008:
12007:
11980:
11979:
11967:
11966:
11930:
11929:
11910:
11908:
11907:
11902:
11891:
11890:
11871:
11869:
11868:
11863:
11852:
11851:
11830:
11829:
11802:
11801:
11789:
11788:
11731:
11730:
11711:
11709:
11708:
11703:
11685:
11683:
11682:
11677:
11675:
11674:
11658:
11656:
11655:
11650:
11648:
11647:
11631:
11629:
11628:
11623:
11611:
11609:
11608:
11603:
11577:
11575:
11574:
11569:
11564:
11563:
11551:
11536:
11534:
11533:
11528:
11517:
11516:
11504:
11489:
11487:
11486:
11481:
11476:
11475:
11463:
11462:
11450:
11436:
11435:
11423:
11405:
11403:
11402:
11397:
11386:
11385:
11370:
11369:
11357:
11356:
11335:
11334:
11291:
11289:
11288:
11283:
11281:
11280:
11264:
11262:
11261:
11256:
11248:
11240:
11225:
11223:
11222:
11217:
11206:
11205:
11193:
11192:
11180:
11179:
11139:
11137:
11136:
11131:
11120:
11119:
11107:
11092:
11090:
11089:
11084:
11072:
11070:
11069:
11064:
11059:
11058:
11046:
11032:
11031:
11019:
11018:
11006:
10988:
10986:
10985:
10980:
10969:
10968:
10953:
10952:
10940:
10939:
10887:
10885:
10884:
10879:
10877:
10865:
10863:
10862:
10857:
10852:
10851:
10826:
10824:
10823:
10818:
10807:
10806:
10794:
10793:
10753:
10751:
10750:
10745:
10743:
10742:
10730:
10729:
10713:
10711:
10710:
10705:
10691:
10679:
10677:
10676:
10671:
10659:
10657:
10656:
10651:
10649:
10648:
10632:
10630:
10629:
10624:
10612:
10610:
10609:
10604:
10602:
10601:
10563:
10561:
10560:
10555:
10547:
10546:
10507:. To check that
10506:
10504:
10503:
10498:
10493:
10492:
10480:
10466:which maximizes
10465:
10463:
10462:
10457:
10445:
10443:
10442:
10437:
10425:
10423:
10422:
10417:
10415:
10414:
10398:
10396:
10395:
10390:
10378:
10376:
10375:
10370:
10358:
10356:
10355:
10350:
10338:
10336:
10335:
10330:
10328:
10327:
10311:
10309:
10308:
10303:
10279:
10277:
10276:
10271:
10259:
10257:
10256:
10251:
10240:
10223:
10211:
10209:
10208:
10203:
10195:
10187:
10175:
10173:
10172:
10167:
10155:
10153:
10152:
10147:
10135:
10133:
10132:
10127:
10125:
10117:
10106:
10089:
10077:
10075:
10074:
10069:
10051:
10049:
10048:
10043:
10035:
10027:
10015:
10013:
10012:
10007:
9992:
9990:
9989:
9984:
9968:
9966:
9965:
9960:
9952:
9951:
9939:
9938:
9898:
9896:
9895:
9890:
9876:
9855:
9853:
9852:
9847:
9829:
9827:
9826:
9821:
9816:
9785:
9783:
9782:
9777:
9772:
9771:
9759:
9744:
9742:
9741:
9736:
9728:
9727:
9715:
9714:
9692:
9690:
9689:
9684:
9666:
9664:
9663:
9658:
9653:
9648:
9647:
9620:
9619:
9599:
9597:
9596:
9591:
9579:
9577:
9576:
9571:
9556:
9554:
9553:
9548:
9546:
9545:
9521:
9476:
9474:
9473:
9468:
9450:
9448:
9447:
9442:
9440:
9432:
9421:
9416:
9415:
9391:
9379:
9377:
9376:
9371:
9363:
9355:
9344:is univalent in
9340:
9338:
9337:
9332:
9324:
9323:
9311:
9310:
9298:
9297:
9257:
9255:
9254:
9249:
9241:
9233:
9221:
9219:
9218:
9213:
9198:
9196:
9195:
9190:
9182:
9181:
9118:Menahem Schiffer
9094:Herbert Grötzsch
9080:
9074:
9072:
9071:
9066:
9054:
9052:
9051:
9046:
9025:
9023:
9022:
9017:
9012:
9011:
9002:
9001:
8980:
8965:
8963:
8962:
8957:
8942:
8940:
8939:
8934:
8926:
8925:
8916:
8915:
8900:
8899:
8887:
8886:
8849:
8847:
8846:
8841:
8829:
8827:
8826:
8821:
8809:
8807:
8806:
8801:
8789:
8787:
8786:
8781:
8767:
8755:
8753:
8752:
8747:
8735:
8729:
8727:
8726:
8721:
8705:
8703:
8702:
8697:
8677:
8675:
8674:
8669:
8657:
8655:
8654:
8649:
8647:
8635:
8633:
8632:
8627:
8622:
8614:
8600:
8560:
8558:
8557:
8552:
8540:
8538:
8537:
8532:
8520:
8518:
8517:
8512:
8496:
8494:
8493:
8488:
8468:
8454:
8449:
8445:
8444:
8442:
8441:
8433:
8424:
8419:
8414:
8412:
8404:
8399:
8383:
8369:
8368:
8363:
8345:
8331:
8317:
8297:
8280:
8278:
8277:
8272:
8270:
8269:
8250:
8248:
8247:
8242:
8240:
8239:
8221:
8207:
8199:
8194:
8180:
8161:
8159:
8158:
8153:
8148:
8146:
8133:
8128:
8114:
8105:
8071:
8070:
8057:
8031:
8029:
8028:
8023:
8011:
8009:
8008:
8003:
7991:
7989:
7988:
7983:
7971:
7969:
7968:
7963:
7951:
7949:
7948:
7943:
7926:
7918:
7907:
7871:
7869:
7868:
7863:
7851:
7849:
7848:
7843:
7816:
7814:
7813:
7808:
7790:
7788:
7787:
7782:
7768:
7766:
7765:
7760:
7748:
7746:
7745:
7740:
7728:
7726:
7725:
7720:
7718:
7717:
7716:
7696:
7694:
7693:
7688:
7671:
7656:
7654:
7653:
7648:
7636:
7634:
7633:
7628:
7611:
7596:
7594:
7593:
7588:
7561:
7559:
7558:
7553:
7541:
7539:
7538:
7533:
7521:
7519:
7518:
7513:
7501:
7499:
7498:
7493:
7491:
7490:
7474:
7472:
7471:
7466:
7454:
7452:
7451:
7446:
7431:
7415:
7413:
7412:
7407:
7405:
7404:
7395:
7394:
7378:
7376:
7375:
7370:
7368:
7367:
7348:
7346:
7345:
7340:
7329:
7314:
7312:
7311:
7306:
7278:
7271:Ahlfors function
7261:. The method of
7256:
7254:
7253:
7248:
7246:
7245:
7232:
7230:
7229:
7224:
7222:
7221:
7208:
7206:
7205:
7200:
7182:
7180:
7179:
7174:
7148:
7133:
7131:
7130:
7125:
7089:
7087:
7086:
7081:
7069:
7067:
7066:
7061:
7050:
7038:
7036:
7035:
7030:
7018:
7016:
7015:
7010:
6989:
6987:
6986:
6981:
6966:
6964:
6963:
6958:
6940:
6938:
6937:
6932:
6911:
6909:
6908:
6903:
6891:
6889:
6888:
6883:
6862:
6860:
6859:
6854:
6846:
6845:
6833:
6832:
6816:
6814:
6813:
6808:
6803:
6802:
6778:
6777:
6755:
6753:
6752:
6747:
6735:
6733:
6732:
6727:
6725:
6714:
6690:
6688:
6687:
6682:
6674:
6656:
6654:
6653:
6648:
6631:
6616:
6614:
6613:
6608:
6581:
6579:
6578:
6573:
6561:
6559:
6558:
6553:
6541:
6539:
6538:
6533:
6521:
6519:
6518:
6513:
6511:
6510:
6494:
6492:
6491:
6486:
6468:
6466:
6465:
6460:
6448:
6446:
6445:
6440:
6432:
6424:
6408:
6406:
6405:
6400:
6392:
6384:
6373:
6353:
6352:
6309:
6307:
6306:
6301:
6284:
6269:
6267:
6266:
6261:
6234:
6232:
6231:
6226:
6224:
6223:
6192:
6190:
6189:
6184:
6172:
6170:
6169:
6164:
6145:
6143:
6142:
6137:
6120:
6105:
6103:
6102:
6097:
6070:
6068:
6067:
6062:
6050:
6048:
6047:
6042:
6030:
6028:
6027:
6022:
6010:
6008:
6007:
6002:
5984:
5982:
5981:
5976:
5974:
5946:
5944:
5943:
5938:
5926:
5924:
5923:
5918:
5913:
5912:
5889:
5887:
5886:
5881:
5855:
5841:
5840:
5825:
5824:
5812:
5811:
5802:
5794:
5786:
5785:
5773:
5772:
5757:
5756:
5744:
5743:
5734:
5726:
5718:
5717:
5705:
5704:
5683:
5682:
5673:
5665:
5651:
5650:
5629:
5628:
5619:
5605:
5603:
5602:
5597:
5595:
5594:
5572:
5570:
5569:
5564:
5552:
5550:
5549:
5544:
5532:
5530:
5529:
5524:
5513:converges, take
5512:
5510:
5509:
5504:
5499:
5498:
5486:
5485:
5469:
5467:
5466:
5461:
5459:
5458:
5442:
5440:
5439:
5434:
5432:
5431:
5415:
5413:
5412:
5407:
5395:
5393:
5392:
5387:
5373:
5359:
5348:
5334:
5333:
5312:
5311:
5302:
5285:
5283:
5282:
5277:
5258:
5253:
5243:
5238:
5217:
5216:
5195:
5194:
5176:
5174:
5173:
5168:
5156:
5154:
5153:
5148:
5136:
5134:
5133:
5128:
5110:
5108:
5107:
5102:
5100:
5099:
5083:
5081:
5080:
5075:
5057:
5055:
5054:
5049:
5037:
5035:
5034:
5029:
5021:
5012:
5003:
4991:
4989:
4988:
4983:
4974:
4955:
4953:
4952:
4947:
4935:
4933:
4932:
4927:
4915:
4913:
4912:
4907:
4889:
4887:
4886:
4881:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4829:
4827:
4826:
4821:
4819:
4818:
4802:
4800:
4799:
4794:
4792:
4791:
4771:
4769:
4768:
4763:
4751:
4749:
4748:
4743:
4741:
4740:
4724:
4722:
4721:
4716:
4714:
4713:
4690:
4688:
4687:
4682:
4667:Montel's theorem
4652:
4650:
4649:
4644:
4642:
4641:
4628:
4626:
4625:
4620:
4618:
4617:
4604:
4602:
4601:
4596:
4584:
4582:
4581:
4576:
4564:
4562:
4561:
4556:
4554:
4553:
4540:
4538:
4537:
4532:
4530:
4529:
4509:
4507:
4506:
4501:
4499:
4498:
4485:
4483:
4482:
4477:
4475:
4474:
4457:
4455:
4454:
4449:
4447:
4446:
4424:
4422:
4421:
4416:
4404:
4402:
4401:
4396:
4384:
4382:
4381:
4376:
4325:
4323:
4322:
4317:
4306:
4305:
4284:
4283:
4262:
4261:
4245:
4243:
4242:
4237:
4201:
4199:
4198:
4193:
4181:
4179:
4178:
4173:
4168:
4154:
4137:
4136:
4124:
4123:
4114:
4112:
4098:
4063:Morera's theorem
4041:
4039:
4038:
4033:
4006:
4004:
4003:
3998:
3986:
3984:
3983:
3978:
3966:
3964:
3963:
3958:
3943:
3941:
3940:
3935:
3923:
3921:
3920:
3915:
3900:
3898:
3897:
3892:
3880:
3878:
3877:
3872:
3860:
3858:
3857:
3852:
3840:
3838:
3837:
3832:
3820:
3818:
3817:
3812:
3800:
3798:
3797:
3792:
3780:
3778:
3777:
3772:
3760:
3758:
3757:
3752:
3740:
3738:
3737:
3732:
3717:
3715:
3714:
3709:
3707:
3706:
3690:
3688:
3687:
3682:
3680:
3679:
3663:
3661:
3660:
3655:
3643:
3641:
3640:
3635:
3623:
3621:
3620:
3615:
3607:
3606:
3590:
3588:
3587:
3582:
3564:
3562:
3561:
3556:
3542:
3541:
3529:
3528:
3512:
3510:
3509:
3504:
3502:
3501:
3485:
3483:
3482:
3477:
3469:
3468:
3452:
3450:
3449:
3444:
3442:
3441:
3421:
3419:
3418:
3413:
3411:
3410:
3394:
3392:
3391:
3386:
3371:
3369:
3368:
3363:
3361:
3360:
3344:
3342:
3341:
3336:
3324:
3322:
3321:
3316:
3304:
3302:
3301:
3296:
3294:
3293:
3277:
3275:
3274:
3269:
3247:
3245:
3244:
3239:
3231:
3230:
3214:
3212:
3211:
3206:
3191:
3189:
3188:
3183:
3171:
3169:
3168:
3163:
3161:
3160:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3104:
3102:
3101:
3096:
3094:
3093:
3077:
3075:
3074:
3069:
3057:
3055:
3054:
3049:
3037:
3035:
3034:
3029:
3027:
3026:
3010:
3008:
3007:
3002:
2985:
2974:
2969:
2968:
2956:
2954:
2943:
2934:
2932:
2931:
2926:
2914:
2912:
2911:
2906:
2904:
2903:
2887:
2885:
2884:
2879:
2871:
2870:
2854:
2852:
2851:
2846:
2834:
2832:
2831:
2826:
2814:
2812:
2811:
2806:
2794:
2792:
2791:
2786:
2771:
2769:
2768:
2763:
2739:
2737:
2736:
2731:
2719:
2717:
2716:
2711:
2706:
2688:
2686:
2685:
2680:
2668:
2666:
2665:
2660:
2648:
2646:
2645:
2640:
2628:
2626:
2625:
2620:
2615:
2600:
2598:
2597:
2592:
2590:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2512:
2510:
2509:
2504:
2492:
2490:
2489:
2484:
2466:
2464:
2463:
2458:
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2398:
2383:
2381:
2380:
2375:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2333:
2332:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2256:
2254:
2253:
2248:
2236:
2234:
2233:
2228:
2226:
2225:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2173:
2172:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2116:
2114:
2113:
2108:
2090:
2088:
2087:
2082:
2080:
2079:
2063:
2061:
2060:
2055:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
2004:
1953:
1951:
1950:
1945:
1933:
1931:
1930:
1925:
1913:
1911:
1910:
1905:
1893:
1891:
1890:
1885:
1880:
1875:
1867:
1861:
1860:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1798:
1796:
1795:
1790:
1775:
1773:
1772:
1767:
1755:
1753:
1752:
1747:
1742:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1676:
1674:
1673:
1668:
1654:
1642:
1640:
1639:
1634:
1619:
1617:
1616:
1611:
1599:
1597:
1596:
1591:
1579:
1577:
1576:
1571:
1555:
1553:
1552:
1547:
1526:
1524:
1523:
1518:
1506:
1504:
1503:
1498:
1483:
1481:
1480:
1475:
1463:
1461:
1460:
1455:
1440:
1438:
1437:
1432:
1417:
1415:
1414:
1409:
1397:
1395:
1394:
1389:
1374:
1372:
1371:
1366:
1351:
1349:
1348:
1343:
1341:
1297:), the ball and
1296:
1294:
1293:
1288:
1270:
1268:
1267:
1262:
1260:
1259:
1254:
1197:
1195:
1194:
1189:
1178:
1170:
1143:
1141:
1140:
1135:
1124:
1116:
1085:
1083:
1082:
1077:
1053:
1051:
1050:
1045:
1034:
1026:
994:angle-preserving
940:Riemann surfaces
929:potential theory
920:
918:
917:
912:
910:
897:Green's function
883:
881:
880:
875:
857:Karl Weierstrass
844:Bernhard Riemann
841:
839:
838:
833:
798:
796:
795:
790:
778:
776:
775:
770:
768:
767:
751:
749:
748:
743:
727:
725:
724:
719:
708:
707:
681:
679:
678:
673:
661:
659:
658:
653:
641:
639:
638:
633:
631:
630:
614:
612:
611:
606:
584:
582:
581:
576:
564:
562:
561:
556:
544:
542:
541:
536:
514:
512:
511:
506:
492:
484:
476:
443:
441:
440:
435:
416:
414:
413:
408:
392:
390:
389:
384:
382:
370:
368:
367:
362:
360:
341:simply connected
335:
333:
332:
327:
310:complex analysis
298:
291:
284:
268:
267:
252:Karl Weierstrass
247:Bernhard Riemann
237:Jacques Hadamard
66:Imaginary number
46:
36:Complex analysis
30:
28:Complex analysis
19:
18:
16969:
16968:
16964:
16963:
16962:
16960:
16959:
16958:
16939:
16938:
16937:
16932:
16912:
16823:Riemann surface
16798:Riemann problem
16714:
16709:
16659:
16626:10.2307/2318448
16600:Springer-Verlag
16588:
16578:Springer-Verlag
16574:Normal families
16556:
16555:
16550:
16535:
16525:Springer-Verlag
16497:de Possel, René
16466:10.2307/1986285
16437:
16385:
16360:
16339:Springer-Verlag
16327:
16317:Springer-Verlag
16305:
16212:
16190:
16169:
16148:
16127:
16117:Springer-Verlag
16109:Conway, John B.
16102:
16092:Springer-Verlag
16084:Conway, John B.
16045:
16035:Springer-Verlag
16023:
16005:
15984:
15965:
15960:
15911:
15907:
15891:10.1.1.100.2423
15874:
15870:
15861:
15857:
15850:
15846:
15838:
15834:
15826:
15822:
15814:
15810:
15802:
15798:
15791:
15787:
15779:
15775:
15768:
15764:
15757:
15753:
15746:
15742:
15735:
15731:
15723:
15719:
15711:
15707:
15700:
15696:
15688:
15684:
15677:
15673:
15666:
15662:
15655:
15651:
15643:
15639:
15632:
15628:
15621:
15617:
15610:
15606:
15599:
15595:
15588:
15584:
15577:
15573:
15565:
15561:
15553:
15549:
15522:
15518:
15510:
15506:
15483:
15479:
15471:
15467:
15466:
15462:
15454:
15450:
15442:
15438:
15427:
15423:
15394:
15390:
15376:
15372:
15363:
15359:
15355:
15332:
15301:
15287:
15284:
15283:
15263:
15259:
15244:
15243:
15241:
15238:
15237:
15216:
15210:
15209:
15208:
15206:
15203:
15202:
15179:
15175:
15160:
15159:
15157:
15154:
15153:
15136:
15132:
15127:
15122:
15119:
15118:
15098:
15094:
15086:
15083:
15082:
15063:
15058:
15055:
15054:
15038:
15035:
15034:
15015:
15012:
15011:
14996:
14958:
14955:
14954:
14932:
14929:
14928:
14912:
14909:
14908:
14889:
14881:
14878:
14877:
14861:
14858:
14857:
14824:
14810:
14793:
14791:
14788:
14787:
14764:
14762:
14759:
14758:
14757:with precision
14733:
14730:
14729:
14723:
14706:
14702:
14694:
14691:
14690:
14665:
14662:
14661:
14639:
14628:
14625:
14624:
14586:
14582:
14565:
14562:
14561:
14541:
14537:
14529:
14526:
14525:
14502:
14500:
14497:
14496:
14477:
14474:
14473:
14454:
14450:
14442:
14439:
14438:
14412:
14408:
14406:
14403:
14402:
14386:
14383:
14382:
14353:
14349:
14342:
14338:
14336:
14333:
14332:
14300:
14296:
14288:
14285:
14284:
14259:
14255:
14248:
14244:
14227:
14224:
14223:
14201:
14198:
14197:
14180:
14176:
14174:
14171:
14170:
14151:
14148:
14147:
14115:
14111:
14109:
14106:
14105:
14086:
14083:
14082:
14059:
14055:
14041:
14024:
14022:
14019:
14018:
13998:
13994:
13992:
13989:
13988:
13987:with precision
13963:
13960:
13959:
13927:
13923:
13915:
13912:
13911:
13895:
13892:
13891:
13875:
13872:
13871:
13845:
13841:
13839:
13836:
13835:
13818:
13814:
13809:
13806:
13805:
13785:
13781:
13779:
13776:
13775:
13774:with precision
13737:
13733:
13716:
13713:
13712:
13695:
13691:
13682:
13678:
13676:
13673:
13672:
13653:
13650:
13649:
13626:
13622:
13620:
13617:
13616:
13600:
13597:
13596:
13572:
13555:
13551:
13549:
13546:
13545:
13518:
13514:
13499:
13495:
13493:
13490:
13489:
13473:
13470:
13469:
13452:
13448:
13433:
13429:
13427:
13424:
13423:
13417:
13396:
13371:
13368:
13367:
13348:
13346:
13343:
13342:
13319:
13316:
13315:
13312:Riemann surface
13295:
13292:
13291:
13284:
13263:
13260:
13259:
13243:
13240:
13239:
13223:
13220:
13219:
13203:
13200:
13199:
13183:
13180:
13179:
13163:
13160:
13159:
13139:
13136:
13135:
13115:
13112:
13111:
13095:
13092:
13091:
13064:
13061:
13060:
13044:
13041:
13040:
13021:
13015:
13011:
13000:
12974:
12971:
12970:
12942:
12939:
12938:
12916:
12899:
12897:
12894:
12893:
12877:
12874:
12873:
12856:
12852:
12850:
12847:
12846:
12830:
12827:
12826:
12810:
12807:
12806:
12775:
12772:
12771:
12739:
12735:
12726:
12722:
12696:
12693:
12692:
12669:
12666:
12665:
12648:
12644:
12642:
12639:
12638:
12622:
12619:
12618:
12602:
12599:
12598:
12575:
12571:
12569:
12566:
12565:
12549:
12546:
12545:
12542:
12516:
12512:
12510:
12507:
12506:
12489:
12485:
12483:
12480:
12479:
12463:
12460:
12459:
12443:
12440:
12439:
12423:
12420:
12419:
12387:
12383:
12381:
12378:
12377:
12357:
12354:
12353:
12333:
12329:
12320:
12316:
12314:
12311:
12310:
12293:
12289:
12287:
12284:
12283:
12251:
12247:
12245:
12242:
12241:
12225:
12222:
12221:
12195:
12191:
12173:
12169:
12151:
12147:
12145:
12142:
12141:
12135:
12118:
12114:
12112:
12109:
12108:
12091:
12087:
12085:
12082:
12081:
12064:
12060:
12058:
12055:
12054:
12025:
12021:
12003:
11999:
11975:
11971:
11959:
11955:
11925:
11921:
11919:
11916:
11915:
11886:
11882:
11880:
11877:
11876:
11847:
11843:
11825:
11821:
11797:
11793:
11781:
11777:
11726:
11722:
11720:
11717:
11716:
11691:
11688:
11687:
11670:
11666:
11664:
11661:
11660:
11643:
11639:
11637:
11634:
11633:
11617:
11614:
11613:
11597:
11594:
11593:
11559:
11555:
11544:
11542:
11539:
11538:
11512:
11508:
11497:
11495:
11492:
11491:
11471:
11467:
11458:
11454:
11443:
11431:
11427:
11416:
11414:
11411:
11410:
11378:
11374:
11365:
11361:
11352:
11348:
11330:
11326:
11300:
11297:
11296:
11276:
11272:
11270:
11267:
11266:
11244:
11236:
11234:
11231:
11230:
11198:
11194:
11188:
11184:
11175:
11171:
11148:
11145:
11144:
11115:
11111:
11100:
11098:
11095:
11094:
11078:
11075:
11074:
11054:
11050:
11039:
11027:
11023:
11014:
11010:
10999:
10997:
10994:
10993:
10961:
10957:
10948:
10944:
10935:
10931:
10896:
10893:
10892:
10873:
10871:
10868:
10867:
10847:
10843:
10835:
10832:
10831:
10799:
10795:
10789:
10785:
10762:
10759:
10758:
10738:
10734:
10725:
10721:
10719:
10716:
10715:
10687:
10685:
10682:
10681:
10665:
10662:
10661:
10644:
10640:
10638:
10635:
10634:
10618:
10615:
10614:
10597:
10593:
10576:
10573:
10572:
10542:
10538:
10515:
10512:
10511:
10488:
10484:
10473:
10471:
10468:
10467:
10451:
10448:
10447:
10431:
10428:
10427:
10410:
10406:
10404:
10401:
10400:
10384:
10381:
10380:
10364:
10361:
10360:
10344:
10341:
10340:
10323:
10319:
10317:
10314:
10313:
10285:
10282:
10281:
10265:
10262:
10261:
10236:
10219:
10217:
10214:
10213:
10191:
10183:
10181:
10178:
10177:
10161:
10158:
10157:
10141:
10138:
10137:
10136:imply that, if
10121:
10113:
10102:
10085:
10083:
10080:
10079:
10057:
10054:
10053:
10031:
10023:
10021:
10018:
10017:
9998:
9995:
9994:
9978:
9975:
9974:
9944:
9940:
9934:
9930:
9907:
9904:
9903:
9872:
9864:
9861:
9860:
9835:
9832:
9831:
9812:
9795:
9792:
9791:
9767:
9763:
9752:
9750:
9747:
9746:
9745:that maximises
9720:
9716:
9710:
9706:
9698:
9695:
9694:
9672:
9669:
9668:
9649:
9643:
9639:
9615:
9611:
9609:
9606:
9605:
9585:
9582:
9581:
9565:
9562:
9561:
9538:
9534:
9517:
9485:
9482:
9481:
9456:
9453:
9452:
9436:
9428:
9417:
9411:
9407:
9387:
9385:
9382:
9381:
9359:
9351:
9349:
9346:
9345:
9316:
9312:
9306:
9302:
9293:
9289:
9266:
9263:
9262:
9237:
9229:
9227:
9224:
9223:
9207:
9204:
9203:
9177:
9173:
9147:
9144:
9143:
9110:extremal metric
9076:
9060:
9057:
9056:
9031:
9028:
9027:
9007:
9003:
8988:
8984:
8973:
8971:
8968:
8967:
8951:
8948:
8947:
8921:
8917:
8911:
8907:
8895:
8891:
8879:
8875:
8858:
8855:
8854:
8835:
8832:
8831:
8815:
8812:
8811:
8795:
8792:
8791:
8763:
8761:
8758:
8757:
8756:is a domain in
8741:
8738:
8737:
8736:-axis. Thus if
8731:
8715:
8712:
8711:
8691:
8688:
8687:
8684:
8663:
8660:
8659:
8643:
8641:
8638:
8637:
8618:
8610:
8596:
8576:
8573:
8572:
8546:
8543:
8542:
8526:
8523:
8522:
8506:
8503:
8502:
8461:
8450:
8434:
8429:
8428:
8420:
8418:
8408:
8400:
8398:
8397:
8393:
8376:
8364:
8359:
8358:
8341:
8327:
8310:
8290:
8288:
8285:
8284:
8265:
8264:
8256:
8253:
8252:
8229:
8225:
8217:
8200:
8195:
8190:
8173:
8171:
8168:
8167:
8115:
8113:
8106:
8063:
8059:
8058:
8056:
8039:
8036:
8035:
8017:
8014:
8013:
7997:
7994:
7993:
7977:
7974:
7973:
7972:. The function
7957:
7954:
7953:
7917:
7903:
7877:
7874:
7873:
7857:
7854:
7853:
7822:
7819:
7818:
7796:
7793:
7792:
7791:. If not, take
7776:
7773:
7772:
7754:
7751:
7750:
7734:
7731:
7730:
7712:
7711:
7704:
7702:
7699:
7698:
7664:
7662:
7659:
7658:
7642:
7639:
7638:
7604:
7602:
7599:
7598:
7567:
7564:
7563:
7547:
7544:
7543:
7527:
7524:
7523:
7507:
7504:
7503:
7486:
7482:
7480:
7477:
7476:
7460:
7457:
7456:
7427:
7421:
7418:
7417:
7400:
7399:
7390:
7386:
7384:
7381:
7380:
7363:
7362:
7354:
7351:
7350:
7322:
7320:
7317:
7316:
7288:
7285:
7284:
7274:
7241:
7240:
7238:
7235:
7234:
7217:
7216:
7214:
7211:
7210:
7188:
7185:
7184:
7141:
7139:
7136:
7135:
7095:
7092:
7091:
7075:
7072:
7071:
7046:
7044:
7041:
7040:
7024:
7021:
7020:
6995:
6992:
6991:
6972:
6969:
6968:
6967:, no points of
6946:
6943:
6942:
6917:
6914:
6913:
6897:
6894:
6893:
6868:
6865:
6864:
6841:
6837:
6828:
6824:
6822:
6819:
6818:
6798:
6794:
6773:
6769:
6761:
6758:
6757:
6741:
6738:
6737:
6713:
6696:
6693:
6692:
6670:
6662:
6659:
6658:
6624:
6622:
6619:
6618:
6587:
6584:
6583:
6567:
6564:
6563:
6547:
6544:
6543:
6527:
6524:
6523:
6506:
6505:
6503:
6500:
6499:
6474:
6471:
6470:
6454:
6451:
6450:
6428:
6420:
6418:
6415:
6414:
6383:
6369:
6345:
6341:
6324:
6321:
6320:
6277:
6275:
6272:
6271:
6240:
6237:
6236:
6216:
6212:
6198:
6195:
6194:
6178:
6175:
6174:
6158:
6155:
6154:
6113:
6111:
6108:
6107:
6076:
6073:
6072:
6056:
6053:
6052:
6036:
6033:
6032:
6016:
6013:
6012:
5990:
5987:
5986:
5970:
5962:
5959:
5958:
5932:
5929:
5928:
5908:
5904:
5899:
5896:
5895:
5851:
5836:
5832:
5820:
5816:
5807:
5803:
5798:
5790:
5781:
5777:
5768:
5764:
5752:
5748:
5739:
5735:
5730:
5722:
5713:
5709:
5700:
5696:
5678:
5674:
5669:
5661:
5646:
5642:
5624:
5620:
5615:
5613:
5610:
5609:
5590:
5586:
5578:
5575:
5574:
5558:
5555:
5554:
5538:
5535:
5534:
5518:
5515:
5514:
5494:
5490:
5481:
5477:
5475:
5472:
5471:
5454:
5450:
5448:
5445:
5444:
5427:
5423:
5421:
5418:
5417:
5401:
5398:
5397:
5396:. Now for each
5369:
5355:
5344:
5329:
5325:
5307:
5303:
5298:
5296:
5293:
5292:
5254:
5249:
5239:
5234:
5212:
5208:
5190:
5186:
5184:
5181:
5180:
5162:
5159:
5158:
5142:
5139:
5138:
5116:
5113:
5112:
5095:
5091:
5089:
5086:
5085:
5063:
5060:
5059:
5043:
5040:
5039:
5017:
5008:
4999:
4997:
4994:
4993:
4970:
4961:
4958:
4957:
4941:
4938:
4937:
4921:
4918:
4917:
4895:
4892:
4891:
4875:
4872:
4871:
4855:
4852:
4851:
4835:
4832:
4831:
4814:
4810:
4808:
4805:
4804:
4787:
4783:
4781:
4778:
4777:
4757:
4754:
4753:
4736:
4732:
4730:
4727:
4726:
4709:
4705:
4703:
4700:
4699:
4676:
4673:
4672:
4655:locally bounded
4637:
4636:
4634:
4631:
4630:
4613:
4612:
4610:
4607:
4606:
4590:
4587:
4586:
4570:
4567:
4566:
4549:
4548:
4546:
4543:
4542:
4525:
4521:
4519:
4516:
4515:
4494:
4493:
4491:
4488:
4487:
4470:
4469:
4467:
4464:
4463:
4442:
4441:
4439:
4436:
4435:
4410:
4407:
4406:
4390:
4387:
4386:
4331:
4328:
4327:
4301:
4297:
4279:
4275:
4257:
4253:
4251:
4248:
4247:
4207:
4204:
4203:
4187:
4184:
4183:
4164:
4147:
4129:
4125:
4119:
4115:
4102:
4097:
4095:
4092:
4091:
4048:
4012:
4009:
4008:
3992:
3989:
3988:
3972:
3969:
3968:
3949:
3946:
3945:
3929:
3926:
3925:
3906:
3903:
3902:
3886:
3883:
3882:
3866:
3863:
3862:
3846:
3843:
3842:
3826:
3823:
3822:
3806:
3803:
3802:
3786:
3783:
3782:
3766:
3763:
3762:
3746:
3743:
3742:
3723:
3720:
3719:
3702:
3698:
3696:
3693:
3692:
3675:
3671:
3669:
3666:
3665:
3649:
3646:
3645:
3629:
3626:
3625:
3602:
3598:
3596:
3593:
3592:
3570:
3567:
3566:
3537:
3533:
3524:
3520:
3518:
3515:
3514:
3497:
3493:
3491:
3488:
3487:
3464:
3460:
3458:
3455:
3454:
3437:
3433:
3431:
3428:
3427:
3406:
3402:
3400:
3397:
3396:
3377:
3374:
3373:
3356:
3352:
3350:
3347:
3346:
3330:
3327:
3326:
3310:
3307:
3306:
3289:
3285:
3283:
3280:
3279:
3257:
3254:
3253:
3226:
3222:
3220:
3217:
3216:
3200:
3197:
3196:
3177:
3174:
3173:
3156:
3152:
3150:
3147:
3146:
3130:
3127:
3126:
3110:
3107:
3106:
3089:
3085:
3083:
3080:
3079:
3063:
3060:
3059:
3043:
3040:
3039:
3022:
3018:
3016:
3013:
3012:
2975:
2970:
2961:
2957:
2947:
2942:
2940:
2937:
2936:
2920:
2917:
2916:
2899:
2895:
2893:
2890:
2889:
2866:
2862:
2860:
2857:
2856:
2840:
2837:
2836:
2820:
2817:
2816:
2800:
2797:
2796:
2777:
2774:
2773:
2745:
2742:
2741:
2725:
2722:
2721:
2702:
2694:
2691:
2690:
2674:
2671:
2670:
2654:
2651:
2650:
2634:
2631:
2630:
2611:
2606:
2603:
2602:
2586:
2584:
2581:
2580:
2564:
2561:
2560:
2544:
2541:
2540:
2518:
2515:
2514:
2498:
2495:
2494:
2472:
2469:
2468:
2452:
2449:
2448:
2432:
2429:
2428:
2394:
2392:
2389:
2388:
2369:
2366:
2365:
2349:
2346:
2345:
2328:
2324:
2322:
2319:
2318:
2302:
2299:
2298:
2282:
2279:
2278:
2262:
2259:
2258:
2242:
2239:
2238:
2221:
2217:
2215:
2212:
2211:
2195:
2192:
2191:
2168:
2164:
2162:
2159:
2158:
2142:
2139:
2138:
2122:
2119:
2118:
2096:
2093:
2092:
2075:
2071:
2069:
2066:
2065:
2049:
2046:
2045:
2026:
2023:
2022:
2000:
1962:
1959:
1958:
1939:
1936:
1935:
1919:
1916:
1915:
1899:
1896:
1895:
1876:
1871:
1863:
1853:
1849:
1847:
1844:
1843:
1824:
1821:
1820:
1804:
1801:
1800:
1784:
1781:
1780:
1761:
1758:
1757:
1738:
1732:
1729:
1728:
1712:
1709:
1708:
1686:
1683:
1682:
1650:
1648:
1645:
1644:
1628:
1625:
1624:
1605:
1602:
1601:
1585:
1582:
1581:
1565:
1562:
1561:
1535:
1532:
1531:
1512:
1509:
1508:
1492:
1489:
1488:
1469:
1466:
1465:
1449:
1446:
1445:
1426:
1423:
1422:
1403:
1400:
1399:
1383:
1380:
1379:
1360:
1357:
1356:
1337:
1329:
1326:
1325:
1319:
1314:
1312:Normal families
1308:
1276:
1273:
1272:
1255:
1250:
1249:
1247:
1244:
1243:
1174:
1166:
1149:
1146:
1145:
1120:
1112:
1095:
1092:
1091:
1059:
1056:
1055:
1030:
1022:
1005:
1002:
1001:
964:
906:
904:
901:
900:
869:
866:
865:
827:
824:
823:
816:
784:
781:
780:
763:
759:
757:
754:
753:
737:
734:
733:
703:
699:
691:
688:
687:
667:
664:
663:
647:
644:
643:
626:
622:
620:
617:
616:
600:
597:
596:
570:
567:
566:
550:
547:
546:
530:
527:
526:
520:Riemann mapping
488:
480:
472:
455:
452:
451:
429:
426:
425:
402:
399:
398:
378:
376:
373:
372:
356:
354:
351:
350:
321:
318:
317:
316:states that if
302:
262:
172:Residue theorem
147:Local primitive
137:Zeros and poles
52:Complex numbers
22:
17:
12:
11:
5:
16967:
16957:
16956:
16951:
16934:
16933:
16931:
16930:
16917:
16914:
16913:
16911:
16910:
16905:
16900:
16895:
16890:
16885:
16880:
16875:
16870:
16865:
16860:
16855:
16850:
16845:
16840:
16835:
16830:
16825:
16820:
16815:
16813:Riemann sphere
16810:
16808:Riemann solver
16805:
16800:
16795:
16790:
16785:
16780:
16775:
16770:
16765:
16760:
16755:
16750:
16745:
16740:
16735:
16730:
16725:
16719:
16716:
16715:
16708:
16707:
16700:
16693:
16685:
16679:
16678:
16658:
16657:External links
16655:
16654:
16653:
16620:(3): 270–276,
16604:
16591:
16586:
16569:
16538:
16533:
16509:
16493:
16440:
16435:
16415:
16388:
16383:
16363:
16358:
16342:
16330:
16325:
16308:
16303:
16290:
16267:
16228:
16202:
16193:
16188:
16172:
16167:
16151:
16146:
16130:
16125:
16105:
16100:
16080:
16048:
16043:
16026:
16021:
16008:
16003:
15987:
15982:
15964:
15961:
15959:
15958:
15905:
15868:
15855:
15844:
15832:
15820:
15808:
15796:
15785:
15773:
15762:
15751:
15740:
15729:
15717:
15705:
15694:
15682:
15671:
15660:
15649:
15637:
15626:
15615:
15604:
15593:
15582:
15571:
15559:
15547:
15545:
15544:
15539:
15534:
15529:
15516:
15504:
15477:
15460:
15448:
15436:
15421:
15388:
15370:
15356:
15354:
15351:
15350:
15349:
15344:
15338:
15331:
15328:
15327:
15326:
15315:
15309:
15306:
15300:
15297:
15294:
15291:
15271:
15266:
15262:
15258:
15255:
15252:
15219:
15190:
15187:
15182:
15178:
15174:
15171:
15168:
15139:
15135:
15130:
15126:
15106:
15101:
15097:
15093:
15090:
15070:
15066:
15062:
15042:
15022:
15019:
15007:
15006:
14980:
14977:
14974:
14971:
14968:
14965:
14962:
14950:digits of the
14939:
14936:
14916:
14896:
14892:
14888:
14885:
14865:
14850:
14849:
14838:
14832:
14829:
14823:
14820:
14817:
14813:
14809:
14806:
14803:
14800:
14796:
14772:
14769:
14746:
14743:
14740:
14737:
14709:
14705:
14701:
14698:
14678:
14675:
14672:
14669:
14649:
14646:
14642:
14638:
14635:
14632:
14612:
14609:
14606:
14603:
14600:
14597:
14594:
14589:
14585:
14581:
14578:
14575:
14572:
14569:
14549:
14544:
14540:
14536:
14533:
14510:
14507:
14484:
14481:
14462:
14457:
14453:
14449:
14446:
14426:
14423:
14420:
14415:
14411:
14390:
14378:
14377:
14366:
14361:
14356:
14352:
14348:
14345:
14341:
14318:
14315:
14312:
14309:
14306:
14303:
14299:
14295:
14292:
14272:
14267:
14262:
14258:
14254:
14251:
14247:
14243:
14240:
14237:
14234:
14231:
14211:
14208:
14205:
14183:
14179:
14158:
14155:
14135:
14130:
14127:
14124:
14121:
14118:
14114:
14093:
14090:
14070:
14065:
14062:
14058:
14054:
14051:
14048:
14044:
14040:
14037:
14034:
14031:
14027:
14004:
14001:
13997:
13976:
13973:
13970:
13967:
13947:
13944:
13941:
13938:
13935:
13930:
13926:
13922:
13919:
13899:
13879:
13857:
13854:
13851:
13848:
13844:
13821:
13817:
13813:
13791:
13788:
13784:
13763:
13760:
13757:
13754:
13751:
13748:
13745:
13740:
13736:
13732:
13729:
13726:
13723:
13720:
13698:
13694:
13690:
13685:
13681:
13660:
13657:
13637:
13634:
13629:
13625:
13604:
13558:
13554:
13532:
13529:
13526:
13521:
13517:
13513:
13510:
13507:
13502:
13498:
13477:
13455:
13451:
13447:
13444:
13441:
13436:
13432:
13416:
13413:
13399:the theory of
13395:
13392:
13375:
13351:
13336:Riemann sphere
13323:
13299:
13283:
13280:
13267:
13247:
13227:
13207:
13187:
13167:
13143:
13119:
13099:
13068:
13048:
13037:
13036:
13024:
13018:
13014:
13010:
13007:
13003:
12999:
12996:
12993:
12990:
12987:
12984:
12981:
12978:
12966:, so we need
12955:
12952:
12949:
12946:
12926:
12923:
12919:
12915:
12912:
12909:
12906:
12902:
12881:
12859:
12855:
12834:
12814:
12794:
12791:
12788:
12785:
12782:
12779:
12768:
12767:
12756:
12751:
12748:
12745:
12742:
12738:
12734:
12729:
12725:
12721:
12718:
12715:
12712:
12709:
12706:
12703:
12700:
12673:
12651:
12647:
12626:
12606:
12586:
12583:
12578:
12574:
12553:
12541:
12538:
12519:
12515:
12492:
12488:
12467:
12447:
12427:
12407:
12404:
12401:
12398:
12395:
12390:
12386:
12361:
12341:
12336:
12332:
12328:
12323:
12319:
12296:
12292:
12271:
12268:
12265:
12262:
12259:
12254:
12250:
12229:
12209:
12206:
12203:
12198:
12194:
12190:
12187:
12184:
12181:
12176:
12172:
12168:
12165:
12162:
12159:
12154:
12150:
12121:
12117:
12094:
12090:
12067:
12063:
12051:
12050:
12039:
12036:
12033:
12028:
12024:
12020:
12017:
12014:
12011:
12006:
12002:
11998:
11995:
11992:
11989:
11986:
11983:
11978:
11974:
11970:
11965:
11962:
11958:
11954:
11951:
11948:
11945:
11942:
11939:
11936:
11933:
11928:
11924:
11900:
11897:
11894:
11889:
11885:
11873:
11872:
11861:
11858:
11855:
11850:
11846:
11842:
11839:
11836:
11833:
11828:
11824:
11820:
11817:
11814:
11811:
11808:
11805:
11800:
11796:
11792:
11787:
11784:
11780:
11776:
11773:
11770:
11767:
11764:
11761:
11758:
11755:
11752:
11749:
11746:
11743:
11740:
11737:
11734:
11729:
11725:
11701:
11698:
11695:
11673:
11669:
11646:
11642:
11621:
11601:
11587:Grunsky (1978)
11583:Goluzin (1969)
11567:
11562:
11558:
11554:
11550:
11547:
11526:
11523:
11520:
11515:
11511:
11507:
11503:
11500:
11479:
11474:
11470:
11466:
11461:
11457:
11453:
11449:
11446:
11442:
11439:
11434:
11430:
11426:
11422:
11419:
11407:
11406:
11395:
11392:
11389:
11384:
11381:
11377:
11373:
11368:
11364:
11360:
11355:
11351:
11347:
11344:
11341:
11338:
11333:
11329:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11279:
11275:
11254:
11251:
11247:
11243:
11239:
11227:
11226:
11215:
11212:
11209:
11204:
11201:
11197:
11191:
11187:
11183:
11178:
11174:
11170:
11167:
11164:
11161:
11158:
11155:
11152:
11129:
11126:
11123:
11118:
11114:
11110:
11106:
11103:
11082:
11062:
11057:
11053:
11049:
11045:
11042:
11038:
11035:
11030:
11026:
11022:
11017:
11013:
11009:
11005:
11002:
10990:
10989:
10978:
10975:
10972:
10967:
10964:
10960:
10956:
10951:
10947:
10943:
10938:
10934:
10930:
10927:
10924:
10921:
10918:
10915:
10912:
10909:
10906:
10903:
10900:
10876:
10855:
10850:
10846:
10842:
10839:
10828:
10827:
10816:
10813:
10810:
10805:
10802:
10798:
10792:
10788:
10784:
10781:
10778:
10775:
10772:
10769:
10766:
10741:
10737:
10733:
10728:
10724:
10703:
10700:
10697:
10694:
10690:
10669:
10647:
10643:
10622:
10600:
10596:
10592:
10589:
10586:
10583:
10580:
10565:
10564:
10553:
10550:
10545:
10541:
10537:
10534:
10531:
10528:
10525:
10522:
10519:
10496:
10491:
10487:
10483:
10479:
10476:
10455:
10435:
10413:
10409:
10388:
10368:
10348:
10326:
10322:
10301:
10298:
10295:
10292:
10289:
10269:
10249:
10246:
10243:
10239:
10235:
10232:
10229:
10226:
10222:
10201:
10198:
10194:
10190:
10186:
10165:
10145:
10124:
10120:
10116:
10112:
10109:
10105:
10101:
10098:
10095:
10092:
10088:
10067:
10064:
10061:
10041:
10038:
10034:
10030:
10026:
10005:
10002:
9982:
9971:
9970:
9958:
9955:
9950:
9947:
9943:
9937:
9933:
9929:
9926:
9923:
9920:
9917:
9914:
9911:
9888:
9885:
9882:
9879:
9875:
9871:
9868:
9845:
9842:
9839:
9819:
9815:
9811:
9808:
9805:
9802:
9799:
9775:
9770:
9766:
9762:
9758:
9755:
9734:
9731:
9726:
9723:
9719:
9713:
9709:
9705:
9702:
9682:
9679:
9676:
9656:
9652:
9646:
9642:
9638:
9635:
9632:
9629:
9626:
9623:
9618:
9614:
9589:
9569:
9558:
9557:
9544:
9541:
9537:
9533:
9530:
9527:
9524:
9520:
9516:
9513:
9510:
9507:
9504:
9501:
9498:
9495:
9492:
9489:
9466:
9463:
9460:
9439:
9435:
9431:
9427:
9424:
9420:
9414:
9410:
9406:
9403:
9400:
9397:
9394:
9390:
9369:
9366:
9362:
9358:
9354:
9342:
9341:
9330:
9327:
9322:
9319:
9315:
9309:
9305:
9301:
9296:
9292:
9288:
9285:
9282:
9279:
9276:
9273:
9270:
9247:
9244:
9240:
9236:
9232:
9211:
9200:
9199:
9188:
9185:
9180:
9176:
9172:
9169:
9166:
9163:
9160:
9157:
9154:
9151:
9098:René de Possel
9090:Jenkins (1958)
9064:
9044:
9041:
9038:
9035:
9015:
9010:
9006:
9000:
8997:
8994:
8991:
8987:
8983:
8979:
8976:
8955:
8944:
8943:
8932:
8929:
8924:
8920:
8914:
8910:
8906:
8903:
8898:
8894:
8890:
8885:
8882:
8878:
8874:
8871:
8868:
8865:
8862:
8839:
8819:
8799:
8779:
8776:
8773:
8770:
8766:
8745:
8719:
8695:
8683:
8680:
8667:
8646:
8625:
8621:
8617:
8613:
8609:
8606:
8603:
8599:
8595:
8592:
8589:
8586:
8583:
8580:
8565:
8564:
8563:
8562:
8550:
8530:
8510:
8499:
8498:
8497:
8486:
8483:
8480:
8477:
8474:
8471:
8467:
8464:
8460:
8457:
8453:
8448:
8440:
8437:
8432:
8427:
8423:
8417:
8411:
8407:
8403:
8396:
8392:
8389:
8386:
8382:
8379:
8375:
8372:
8367:
8362:
8357:
8354:
8351:
8348:
8344:
8340:
8337:
8334:
8330:
8326:
8323:
8320:
8316:
8313:
8309:
8306:
8303:
8300:
8296:
8293:
8268:
8263:
8260:
8238:
8235:
8232:
8228:
8224:
8220:
8216:
8213:
8210:
8206:
8203:
8198:
8193:
8189:
8186:
8183:
8179:
8176:
8164:
8163:
8162:
8151:
8145:
8142:
8139:
8136:
8131:
8127:
8124:
8121:
8118:
8112:
8109:
8104:
8101:
8098:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8074:
8069:
8066:
8062:
8055:
8052:
8049:
8046:
8043:
8021:
8001:
7981:
7961:
7941:
7938:
7935:
7932:
7929:
7924:
7921:
7916:
7913:
7910:
7906:
7902:
7899:
7896:
7893:
7890:
7887:
7884:
7881:
7861:
7841:
7838:
7835:
7832:
7829:
7826:
7806:
7803:
7800:
7780:
7758:
7738:
7715:
7710:
7707:
7686:
7683:
7680:
7677:
7674:
7670:
7667:
7646:
7626:
7623:
7620:
7617:
7614:
7610:
7607:
7586:
7583:
7580:
7577:
7574:
7571:
7551:
7531:
7511:
7489:
7485:
7464:
7444:
7441:
7438:
7434:
7430:
7426:
7403:
7398:
7393:
7389:
7366:
7361:
7358:
7338:
7335:
7332:
7328:
7325:
7304:
7301:
7298:
7295:
7292:
7244:
7220:
7198:
7195:
7192:
7172:
7169:
7166:
7163:
7160:
7157:
7154:
7151:
7147:
7144:
7123:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7099:
7079:
7059:
7056:
7053:
7049:
7028:
7008:
7005:
7002:
6999:
6979:
6976:
6956:
6953:
6950:
6930:
6927:
6924:
6921:
6901:
6881:
6878:
6875:
6872:
6852:
6849:
6844:
6840:
6836:
6831:
6827:
6806:
6801:
6797:
6793:
6790:
6787:
6784:
6781:
6776:
6772:
6768:
6765:
6745:
6723:
6720:
6717:
6712:
6709:
6706:
6703:
6700:
6680:
6677:
6673:
6669:
6666:
6646:
6643:
6640:
6637:
6634:
6630:
6627:
6606:
6603:
6600:
6597:
6594:
6591:
6571:
6551:
6531:
6509:
6496:
6484:
6481:
6478:
6458:
6438:
6435:
6431:
6427:
6423:
6411:
6410:
6409:
6398:
6395:
6390:
6387:
6382:
6379:
6376:
6372:
6368:
6365:
6362:
6359:
6356:
6351:
6348:
6344:
6340:
6337:
6334:
6331:
6328:
6299:
6296:
6293:
6290:
6287:
6283:
6280:
6259:
6256:
6253:
6250:
6247:
6244:
6222:
6219:
6215:
6211:
6208:
6205:
6202:
6182:
6162:
6148:
6147:
6135:
6132:
6129:
6126:
6123:
6119:
6116:
6095:
6092:
6089:
6086:
6083:
6080:
6060:
6040:
6020:
6000:
5997:
5994:
5973:
5969:
5966:
5951:
5950:
5949:
5948:
5936:
5916:
5911:
5907:
5903:
5892:
5891:
5890:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5858:
5854:
5850:
5847:
5844:
5839:
5835:
5831:
5828:
5823:
5819:
5815:
5810:
5806:
5801:
5797:
5793:
5789:
5784:
5780:
5776:
5771:
5767:
5763:
5760:
5755:
5751:
5747:
5742:
5738:
5733:
5729:
5725:
5721:
5716:
5712:
5708:
5703:
5699:
5695:
5692:
5689:
5686:
5681:
5677:
5672:
5668:
5664:
5660:
5657:
5654:
5649:
5645:
5641:
5638:
5635:
5632:
5627:
5623:
5618:
5593:
5589:
5585:
5582:
5562:
5542:
5522:
5502:
5497:
5493:
5489:
5484:
5480:
5457:
5453:
5430:
5426:
5405:
5385:
5382:
5379:
5376:
5372:
5368:
5365:
5362:
5358:
5354:
5351:
5347:
5343:
5340:
5337:
5332:
5328:
5324:
5321:
5318:
5315:
5310:
5306:
5301:
5289:
5288:
5287:
5275:
5272:
5268:
5265:
5262:
5257:
5252:
5248:
5242:
5237:
5233:
5229:
5226:
5223:
5220:
5215:
5211:
5207:
5204:
5201:
5198:
5193:
5189:
5166:
5146:
5126:
5123:
5120:
5098:
5094:
5073:
5070:
5067:
5047:
5027:
5024:
5020:
5015:
5011:
5007:
5002:
4981:
4977:
4973:
4969:
4965:
4945:
4925:
4905:
4902:
4899:
4879:
4859:
4839:
4817:
4813:
4790:
4786:
4761:
4739:
4735:
4712:
4708:
4693:
4692:
4680:
4653:is said to be
4640:
4616:
4594:
4574:
4552:
4528:
4524:
4497:
4473:
4445:
4429:
4428:
4427:
4426:
4414:
4394:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4315:
4312:
4309:
4304:
4300:
4296:
4293:
4290:
4287:
4282:
4278:
4274:
4271:
4268:
4265:
4260:
4256:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4191:
4171:
4167:
4163:
4160:
4157:
4153:
4150:
4146:
4143:
4140:
4135:
4132:
4128:
4122:
4118:
4111:
4108:
4105:
4101:
4085:
4084:
4073:
4072:
4071:
4070:
4056:
4055:
4047:
4044:
4031:
4028:
4025:
4022:
4019:
4016:
3996:
3976:
3956:
3953:
3933:
3913:
3910:
3890:
3870:
3850:
3830:
3810:
3790:
3770:
3750:
3730:
3727:
3705:
3701:
3678:
3674:
3653:
3633:
3613:
3610:
3605:
3601:
3580:
3577:
3574:
3554:
3551:
3548:
3545:
3540:
3536:
3532:
3527:
3523:
3500:
3496:
3475:
3472:
3467:
3463:
3440:
3436:
3409:
3405:
3384:
3381:
3359:
3355:
3334:
3314:
3292:
3288:
3267:
3264:
3261:
3237:
3234:
3229:
3225:
3204:
3181:
3159:
3155:
3134:
3114:
3092:
3088:
3067:
3058:, thus giving
3047:
3025:
3021:
3000:
2997:
2994:
2991:
2988:
2984:
2981:
2978:
2973:
2967:
2964:
2960:
2953:
2950:
2946:
2924:
2902:
2898:
2877:
2874:
2869:
2865:
2844:
2824:
2804:
2795:does not meet
2784:
2781:
2761:
2758:
2755:
2752:
2749:
2729:
2709:
2705:
2701:
2698:
2678:
2658:
2638:
2618:
2614:
2610:
2589:
2568:
2548:
2528:
2525:
2522:
2502:
2482:
2479:
2476:
2456:
2436:
2416:
2413:
2410:
2407:
2404:
2401:
2397:
2373:
2353:
2331:
2327:
2306:
2286:
2266:
2246:
2224:
2220:
2199:
2179:
2176:
2171:
2167:
2146:
2126:
2106:
2103:
2100:
2078:
2074:
2053:
2030:
2010:
2007:
2003:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1943:
1923:
1903:
1883:
1879:
1874:
1870:
1866:
1859:
1856:
1852:
1828:
1808:
1788:
1765:
1745:
1741:
1736:
1716:
1696:
1693:
1690:
1679:
1678:
1666:
1663:
1660:
1657:
1653:
1632:
1621:
1609:
1589:
1569:
1558:winding number
1545:
1542:
1539:
1528:
1516:
1496:
1485:
1473:
1453:
1442:
1430:
1419:
1407:
1387:
1376:
1364:
1340:
1336:
1333:
1318:
1315:
1310:Main article:
1307:
1304:
1303:
1302:
1286:
1283:
1280:
1258:
1253:
1236:
1218:homeomorphisms
1214:
1203:
1187:
1184:
1181:
1177:
1173:
1169:
1165:
1162:
1159:
1156:
1153:
1133:
1130:
1127:
1123:
1119:
1115:
1111:
1108:
1105:
1102:
1099:
1075:
1072:
1069:
1066:
1063:
1043:
1040:
1037:
1033:
1029:
1025:
1021:
1018:
1015:
1012:
1009:
997:
983:differentiable
975:
963:
960:
909:
873:
831:
815:
812:
808:Riemann sphere
788:
766:
762:
741:
717:
714:
711:
706:
702:
698:
695:
671:
651:
629:
625:
604:
593:Henri Poincaré
574:
554:
534:
516:
515:
504:
501:
498:
495:
491:
487:
483:
479:
475:
471:
468:
465:
462:
459:
446:open unit disk
433:
406:
381:
359:
325:
304:
303:
301:
300:
293:
286:
278:
275:
274:
273:
272:
257:
256:
255:
254:
249:
244:
239:
234:
229:
227:Leonhard Euler
224:
216:
215:
209:
208:
202:
201:
200:
199:
194:
189:
184:
179:
174:
169:
164:
162:Laurent series
159:
157:Winding number
154:
149:
144:
139:
131:
130:
124:
123:
122:
121:
116:
111:
106:
101:
93:
92:
86:
85:
84:
83:
78:
73:
68:
63:
55:
54:
48:
47:
39:
38:
32:
31:
15:
9:
6:
4:
3:
2:
16966:
16955:
16952:
16950:
16947:
16946:
16944:
16929:
16928:
16919:
16918:
16915:
16909:
16906:
16904:
16901:
16899:
16896:
16894:
16891:
16889:
16886:
16884:
16881:
16879:
16876:
16874:
16871:
16869:
16866:
16864:
16861:
16859:
16856:
16854:
16851:
16849:
16846:
16844:
16841:
16839:
16836:
16834:
16831:
16829:
16826:
16824:
16821:
16819:
16816:
16814:
16811:
16809:
16806:
16804:
16801:
16799:
16796:
16794:
16791:
16789:
16786:
16784:
16781:
16779:
16776:
16774:
16771:
16769:
16766:
16764:
16761:
16759:
16756:
16754:
16751:
16749:
16746:
16744:
16741:
16739:
16736:
16734:
16731:
16729:
16726:
16724:
16721:
16720:
16717:
16713:
16706:
16701:
16699:
16694:
16692:
16687:
16686:
16683:
16676:
16672:
16671:
16666:
16661:
16660:
16651:
16647:
16643:
16639:
16635:
16631:
16627:
16623:
16619:
16615:
16614:
16609:
16605:
16601:
16597:
16592:
16589:
16583:
16579:
16575:
16570:
16566:
16560:
16549:
16548:
16543:
16539:
16536:
16534:0-387-98221-3
16530:
16526:
16522:
16521:Leslie M. Kay
16518:
16514:
16510:
16506:
16502:
16498:
16494:
16491:
16487:
16483:
16479:
16475:
16471:
16467:
16463:
16459:
16455:
16451:
16450:
16445:
16444:Osgood, W. F.
16441:
16438:
16436:9780486611372
16432:
16428:
16424:
16420:
16416:
16413:
16409:
16405:
16401:
16397:
16393:
16389:
16386:
16384:0-8176-4339-7
16380:
16376:
16372:
16368:
16364:
16361:
16359:9780521809375
16355:
16351:
16347:
16343:
16340:
16336:
16331:
16328:
16322:
16318:
16314:
16309:
16306:
16300:
16296:
16291:
16288:
16284:
16280:
16277:(in German),
16276:
16272:
16268:
16264:
16259:
16254:
16249:
16245:
16241:
16237:
16236:Kim, Kang‑Tae
16233:
16229:
16226:
16222:
16219:(34): 47–94,
16218:
16211:
16207:
16203:
16199:
16194:
16191:
16189:0-387-95069-9
16185:
16181:
16177:
16173:
16170:
16168:0-387-90795-5
16164:
16160:
16156:
16152:
16149:
16147:9780121259501
16143:
16139:
16135:
16131:
16128:
16126:0-387-94460-5
16122:
16118:
16114:
16110:
16106:
16103:
16101:0-387-90328-3
16097:
16093:
16089:
16085:
16081:
16078:
16074:
16070:
16066:
16062:
16058:
16054:
16049:
16046:
16040:
16036:
16032:
16027:
16024:
16022:0-8493-8270-X
16018:
16014:
16009:
16006:
16000:
15996:
15992:
15988:
15985:
15979:
15975:
15971:
15967:
15966:
15954:
15950:
15946:
15942:
15938:
15934:
15929:
15924:
15920:
15916:
15909:
15901:
15897:
15892:
15887:
15883:
15879:
15872:
15865:
15859:
15853:
15848:
15841:
15836:
15829:
15824:
15817:
15812:
15805:
15800:
15794:
15789:
15782:
15777:
15771:
15766:
15760:
15755:
15749:
15744:
15738:
15733:
15726:
15721:
15714:
15709:
15703:
15698:
15691:
15686:
15680:
15675:
15669:
15664:
15658:
15653:
15647:, p. 309
15646:
15641:
15635:
15630:
15624:
15619:
15613:
15608:
15602:
15597:
15591:
15586:
15580:
15575:
15568:
15563:
15556:
15551:
15543:
15540:
15538:
15535:
15533:
15530:
15528:
15525:
15524:
15520:
15513:
15508:
15500:
15496:
15492:
15488:
15481:
15470:
15464:
15457:
15452:
15445:
15440:
15434:
15430:
15425:
15418:
15414:
15410:
15406:
15402:
15398:
15392:
15384:
15380:
15379:Ahlfors, Lars
15374:
15367:
15361:
15357:
15348:
15345:
15342:
15339:
15337:
15334:
15333:
15313:
15307:
15304:
15298:
15295:
15292:
15289:
15264:
15260:
15256:
15253:
15236:reducible to
15235:
15217:
15188:
15180:
15176:
15172:
15169:
15137:
15133:
15128:
15124:
15099:
15095:
15088:
15068:
15064:
15060:
15020:
15009:
15008:
15004:
14999:
14994:
14978:
14972:
14969:
14960:
14953:
14937:
14934:
14914:
14907:and a number
14894:
14890:
14886:
14883:
14855:
14854:
14853:
14836:
14830:
14827:
14821:
14818:
14815:
14804:
14798:
14770:
14767:
14741:
14735:
14726:
14707:
14703:
14699:
14696:
14673:
14667:
14644:
14640:
14636:
14630:
14607:
14604:
14601:
14587:
14583:
14579:
14570:
14567:
14542:
14538:
14531:
14508:
14505:
14460:
14455:
14451:
14447:
14444:
14424:
14418:
14413:
14409:
14380:
14379:
14364:
14354:
14350:
14343:
14339:
14313:
14310:
14307:
14297:
14293:
14290:
14270:
14260:
14256:
14249:
14245:
14241:
14235:
14229:
14209:
14206:
14203:
14181:
14177:
14133:
14125:
14119:
14116:
14112:
14068:
14063:
14060:
14056:
14052:
14049:
14046:
14035:
14029:
14002:
13999:
13995:
13971:
13965:
13945:
13933:
13928:
13924:
13917:
13877:
13852:
13846:
13842:
13819:
13815:
13811:
13789:
13786:
13782:
13758:
13755:
13752:
13738:
13734:
13730:
13721:
13718:
13696:
13692:
13688:
13683:
13679:
13632:
13627:
13623:
13594:
13593:
13592:
13589:
13586:
13584:
13580:
13575:
13556:
13552:
13530:
13527:
13524:
13519:
13515:
13511:
13508:
13505:
13500:
13496:
13475:
13453:
13449:
13445:
13442:
13439:
13434:
13430:
13420:
13412:
13410:
13406:
13402:
13391:
13389:
13373:
13366:
13341:
13340:complex plane
13337:
13321:
13313:
13297:
13289:
13279:
13265:
13245:
13225:
13205:
13185:
13165:
13157:
13141:
13133:
13117:
13097:
13088:
13086:
13082:
13066:
13046:
13016:
13012:
13008:
13005:
12997:
12994:
12991:
12988:
12982:
12976:
12969:
12968:
12967:
12953:
12947:
12944:
12924:
12921:
12910:
12904:
12892:. We require
12879:
12857:
12853:
12832:
12812:
12792:
12789:
12786:
12783:
12780:
12777:
12754:
12746:
12740:
12736:
12727:
12723:
12719:
12716:
12710:
12704:
12698:
12691:
12690:
12689:
12687:
12671:
12649:
12645:
12624:
12604:
12584:
12581:
12576:
12572:
12551:
12537:
12535:
12517:
12513:
12490:
12486:
12478:close to the
12465:
12445:
12438:is zero (any
12425:
12405:
12402:
12396:
12388:
12384:
12375:
12359:
12334:
12330:
12321:
12317:
12294:
12290:
12269:
12266:
12260:
12252:
12248:
12227:
12204:
12196:
12192:
12188:
12182:
12174:
12170:
12166:
12160:
12152:
12148:
12138:
12134:have a fixed
12119:
12115:
12092:
12088:
12065:
12061:
12037:
12034:
12031:
12026:
12018:
12015:
12012:
12004:
12000:
11996:
11990:
11987:
11984:
11976:
11972:
11968:
11963:
11960:
11952:
11949:
11946:
11940:
11934:
11926:
11922:
11914:
11913:
11912:
11895:
11887:
11883:
11859:
11856:
11853:
11848:
11840:
11837:
11834:
11826:
11822:
11818:
11812:
11809:
11806:
11798:
11794:
11790:
11785:
11782:
11774:
11771:
11768:
11762:
11756:
11750:
11747:
11744:
11741:
11735:
11727:
11723:
11715:
11714:
11713:
11699:
11696:
11693:
11671:
11667:
11644:
11640:
11619:
11599:
11592:
11588:
11584:
11579:
11560:
11556:
11524:
11521:
11513:
11509:
11472:
11468:
11464:
11459:
11455:
11440:
11432:
11428:
11393:
11390:
11387:
11382:
11379:
11375:
11366:
11362:
11358:
11353:
11349:
11342:
11339:
11336:
11331:
11327:
11323:
11314:
11308:
11302:
11295:
11294:
11293:
11277:
11273:
11252:
11249:
11241:
11229:mapping from
11213:
11210:
11207:
11202:
11199:
11195:
11189:
11185:
11181:
11176:
11172:
11168:
11165:
11162:
11156:
11150:
11143:
11142:
11141:
11127:
11124:
11116:
11112:
11093:. Therefore,
11080:
11055:
11051:
11036:
11028:
11024:
11020:
11015:
11011:
10976:
10973:
10970:
10965:
10962:
10958:
10949:
10945:
10941:
10936:
10932:
10925:
10922:
10919:
10910:
10904:
10898:
10891:
10890:
10889:
10848:
10844:
10837:
10814:
10811:
10808:
10803:
10800:
10796:
10790:
10786:
10782:
10779:
10776:
10770:
10764:
10757:
10756:
10755:
10739:
10735:
10731:
10726:
10722:
10692:
10667:
10645:
10641:
10620:
10598:
10594:
10590:
10584:
10578:
10570:
10551:
10548:
10543:
10539:
10535:
10532:
10529:
10523:
10517:
10510:
10509:
10508:
10489:
10485:
10453:
10433:
10411:
10407:
10366:
10346:
10324:
10320:
10287:
10267:
10247:
10244:
10241:
10230:
10224:
10199:
10196:
10188:
10163:
10143:
10118:
10110:
10107:
10096:
10090:
10065:
10062:
10059:
10039:
10036:
10028:
10003:
9980:
9956:
9953:
9948:
9945:
9941:
9935:
9931:
9927:
9924:
9921:
9915:
9909:
9902:
9901:
9900:
9877:
9869:
9866:
9857:
9843:
9840:
9837:
9817:
9813:
9806:
9803:
9797:
9789:
9768:
9764:
9732:
9729:
9724:
9721:
9717:
9711:
9707:
9703:
9700:
9680:
9677:
9674:
9654:
9650:
9644:
9640:
9636:
9633:
9630:
9624:
9616:
9612:
9603:
9602:Schwarz lemma
9587:
9567:
9542:
9539:
9531:
9528:
9522:
9518:
9514:
9508:
9502:
9499:
9493:
9487:
9480:
9479:
9478:
9464:
9461:
9458:
9433:
9425:
9422:
9412:
9408:
9404:
9398:
9392:
9367:
9364:
9356:
9328:
9325:
9320:
9317:
9313:
9307:
9303:
9299:
9294:
9290:
9286:
9283:
9280:
9274:
9268:
9261:
9260:
9259:
9245:
9242:
9234:
9209:
9186:
9183:
9178:
9174:
9170:
9167:
9164:
9161:
9155:
9149:
9142:
9141:
9140:
9138:
9134:
9133:Schiff (1993)
9130:
9127:
9123:
9119:
9115:
9111:
9107:
9103:
9099:
9095:
9091:
9087:
9086:David Hilbert
9082:
9079:
9062:
9039:
9033:
9008:
9004:
8998:
8995:
8992:
8989:
8985:
8966:, maximizing
8930:
8927:
8922:
8918:
8912:
8908:
8904:
8901:
8896:
8892:
8888:
8883:
8880:
8876:
8872:
8866:
8860:
8853:
8852:
8851:
8837:
8817:
8768:
8743:
8734:
8717:
8709:
8693:
8679:
8665:
8615:
8607:
8604:
8597:
8593:
8590:
8584:
8578:
8569:
8548:
8528:
8508:
8500:
8484:
8481:
8478:
8472:
8465:
8462:
8458:
8455:
8451:
8446:
8438:
8435:
8425:
8415:
8405:
8394:
8387:
8380:
8377:
8373:
8365:
8352:
8346:
8338:
8335:
8328:
8321:
8314:
8311:
8307:
8301:
8294:
8291:
8283:
8282:
8261:
8258:
8236:
8233:
8230:
8226:
8222:
8211:
8204:
8201:
8191:
8184:
8177:
8174:
8165:
8149:
8140:
8134:
8122:
8116:
8110:
8107:
8096:
8090:
8087:
8081:
8075:
8067:
8064:
8060:
8053:
8047:
8041:
8034:
8033:
8019:
7999:
7979:
7959:
7933:
7927:
7919:
7914:
7911:
7904:
7897:
7894:
7888:
7882:
7859:
7836:
7830:
7824:
7804:
7801:
7798:
7778:
7771:
7756:
7736:
7708:
7705:
7684:
7681:
7675:
7668:
7665:
7644:
7624:
7621:
7615:
7608:
7605:
7584:
7581:
7575:
7569:
7549:
7529:
7509:
7487:
7483:
7462:
7439:
7432:
7428:
7424:
7396:
7391:
7387:
7359:
7356:
7333:
7326:
7323:
7299:
7296:
7293:
7290:
7282:
7277:
7272:
7268:
7265:is to use an
7264:
7260:
7196:
7193:
7190:
7170:
7167:
7158:
7152:
7145:
7142:
7121:
7118:
7109:
7103:
7097:
7077:
7054:
7026:
7003:
6997:
6974:
6954:
6951:
6948:
6925:
6919:
6876:
6870:
6850:
6847:
6842:
6838:
6834:
6829:
6825:
6799:
6795:
6788:
6785:
6782:
6774:
6770:
6763:
6743:
6721:
6718:
6715:
6710:
6704:
6698:
6678:
6667:
6664:
6644:
6641:
6635:
6628:
6625:
6604:
6601:
6595:
6589:
6569:
6549:
6529:
6497:
6482:
6479:
6476:
6456:
6436:
6433:
6425:
6412:
6393:
6385:
6380:
6377:
6370:
6363:
6360:
6357:
6349:
6346:
6342:
6338:
6332:
6326:
6319:
6318:
6317:
6313:
6312:Schwarz lemma
6310:. But by the
6297:
6294:
6288:
6281:
6278:
6257:
6254:
6248:
6242:
6220:
6217:
6213:
6209:
6206:
6203:
6200:
6180:
6160:
6152:
6151:
6150:
6149:
6133:
6130:
6124:
6117:
6114:
6093:
6090:
6084:
6078:
6058:
6038:
6018:
5998:
5995:
5992:
5967:
5964:
5956:
5953:
5952:
5934:
5909:
5905:
5893:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5856:
5845:
5837:
5833:
5829:
5821:
5817:
5808:
5804:
5795:
5782:
5778:
5769:
5765:
5761:
5753:
5749:
5740:
5736:
5727:
5714:
5710:
5701:
5697:
5693:
5687:
5679:
5675:
5666:
5655:
5647:
5643:
5639:
5633:
5625:
5621:
5608:
5607:
5591:
5587:
5583:
5580:
5560:
5540:
5520:
5495:
5491:
5482:
5478:
5455:
5451:
5428:
5424:
5403:
5383:
5380:
5377:
5374:
5366:
5363:
5360:
5352:
5349:
5338:
5330:
5326:
5322:
5316:
5308:
5304:
5291:we have that
5290:
5273:
5270:
5263:
5250:
5246:
5240:
5235:
5231:
5227:
5221:
5213:
5209:
5205:
5199:
5191:
5187:
5179:
5178:
5164:
5144:
5124:
5121:
5118:
5096:
5092:
5071:
5068:
5065:
5045:
5025:
5022:
5013:
5009:
5005:
4975:
4971:
4967:
4943:
4923:
4903:
4900:
4897:
4877:
4857:
4837:
4815:
4811:
4788:
4784:
4775:
4759:
4737:
4733:
4710:
4706:
4697:
4696:
4695:
4694:
4678:
4670:
4668:
4664:
4663:
4662:
4660:
4656:
4605:also lies in
4592:
4572:
4526:
4522:
4513:
4461:
4433:
4412:
4392:
4369:
4363:
4360:
4354:
4348:
4345:
4339:
4333:
4310:
4302:
4298:
4294:
4288:
4280:
4276:
4272:
4266:
4258:
4254:
4230:
4224:
4221:
4215:
4209:
4189:
4169:
4158:
4151:
4148:
4141:
4133:
4130:
4126:
4120:
4116:
4109:
4106:
4103:
4099:
4089:
4088:
4087:
4086:
4081:
4079:
4075:
4074:
4068:
4064:
4060:
4059:
4058:
4057:
4053:
4050:
4049:
4043:
4029:
4026:
4023:
4017:
4014:
3994:
3974:
3954:
3951:
3931:
3911:
3888:
3868:
3848:
3828:
3808:
3788:
3768:
3748:
3728:
3725:
3703:
3699:
3676:
3672:
3651:
3631:
3611:
3608:
3603:
3599:
3578:
3575:
3572:
3552:
3549:
3546:
3543:
3538:
3534:
3530:
3525:
3521:
3498:
3494:
3473:
3470:
3465:
3461:
3438:
3434:
3425:
3407:
3403:
3382:
3379:
3357:
3353:
3332:
3312:
3290:
3286:
3265:
3262:
3259:
3251:
3235:
3232:
3227:
3223:
3202:
3193:
3192:is non-zero.
3179:
3157:
3153:
3132:
3112:
3090:
3086:
3065:
3045:
3023:
3019:
2995:
2992:
2989:
2965:
2958:
2951:
2948:
2944:
2922:
2900:
2896:
2875:
2872:
2867:
2863:
2842:
2822:
2802:
2782:
2756:
2753:
2750:
2747:
2727:
2707:
2703:
2699:
2696:
2676:
2656:
2636:
2629:with a point
2616:
2612:
2608:
2566:
2546:
2526:
2523:
2520:
2513:bounded. Let
2500:
2480:
2477:
2454:
2434:
2414:
2399:
2385:
2371:
2351:
2329:
2325:
2304:
2284:
2264:
2244:
2222:
2218:
2197:
2177:
2174:
2169:
2165:
2144:
2124:
2104:
2101:
2098:
2076:
2072:
2051:
2042:
2028:
2005:
2001:
1994:
1988:
1982:
1979:
1976:
1970:
1964:
1955:
1941:
1921:
1901:
1881:
1872:
1868:
1857:
1854:
1850:
1840:
1826:
1806:
1786:
1777:
1763:
1743:
1734:
1714:
1694:
1691:
1688:
1677:is connected.
1655:
1630:
1622:
1607:
1587:
1567:
1559:
1543:
1540:
1537:
1529:
1514:
1494:
1486:
1471:
1451:
1443:
1428:
1420:
1405:
1385:
1377:
1362:
1355:
1354:
1353:
1334:
1331:
1323:
1313:
1300:
1284:
1281:
1278:
1256:
1241:
1237:
1234:
1230:
1226:
1223:
1219:
1215:
1212:
1208:
1204:
1201:
1182:
1179:
1171:
1163:
1160:
1157:
1154:
1128:
1125:
1117:
1109:
1106:
1103:
1100:
1089:
1073:
1070:
1067:
1064:
1061:
1038:
1035:
1027:
1019:
1016:
1013:
1010:
998:
995:
991:
987:
986:fractal curve
984:
980:
976:
973:
969:
968:
967:
959:
957:
953:
952:Frigyes Riesz
949:
945:
941:
936:
934:
930:
926:
922:
898:
894:
889:
887:
871:
862:
861:David Hilbert
858:
854:
849:
845:
829:
821:
811:
809:
804:
802:
801:Schwarz lemma
786:
764:
760:
752:at the point
739:
731:
715:
712:
704:
700:
693:
685:
669:
649:
627:
623:
602:
594:
590:
588:
587:conformal map
572:
552:
532:
523:
521:
502:
496:
493:
485:
477:
469:
466:
460:
457:
450:
449:
448:
447:
431:
423:
420:
404:
396:
395:biholomorphic
349:
345:
342:
339:
323:
315:
311:
299:
294:
292:
287:
285:
280:
279:
277:
276:
271:
266:
261:
260:
259:
258:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
219:
218:
217:
214:
211:
210:
207:
204:
203:
198:
195:
193:
190:
188:
187:Schwarz lemma
185:
183:
182:Conformal map
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
134:
133:
132:
129:
126:
125:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
96:
95:
94:
91:
88:
87:
82:
79:
77:
74:
72:
71:Complex plane
69:
67:
64:
62:
59:
58:
57:
56:
53:
50:
49:
45:
41:
40:
37:
34:
33:
29:
25:
21:
20:
16925:
16793:Riemann form
16787:
16668:
16617:
16611:
16608:Walsh, J. L.
16595:
16573:
16546:
16516:
16504:
16453:
16447:
16422:
16419:Nehari, Zeev
16403:
16399:
16370:
16349:
16334:
16312:
16294:
16278:
16274:
16243:
16239:
16216:
16206:Gray, Jeremy
16197:
16179:
16158:
16155:Duren, P. L.
16137:
16134:Duren, P. L.
16112:
16087:
16060:
16056:
16030:
16012:
15994:
15973:
15928:math/0505617
15918:
15914:
15908:
15881:
15877:
15871:
15864:Jordan curve
15858:
15847:
15840:Gamelin 2001
15835:
15828:Goluzin 1969
15823:
15811:
15804:Goluzin 1969
15799:
15788:
15781:Goluzin 1969
15776:
15765:
15754:
15743:
15737:Schober 1975
15732:
15725:Jenkins 1958
15720:
15708:
15697:
15690:Jenkins 1958
15685:
15679:Ahlfors 1978
15674:
15663:
15652:
15645:Gamelin 2001
15640:
15629:
15618:
15607:
15596:
15590:Gamelin 2001
15585:
15579:Gamelin 2001
15574:
15562:
15555:Gamelin 2001
15550:
15542:Gamelin 2001
15532:Beardon 1979
15527:Ahlfors 1978
15519:
15512:Remmert 1998
15507:
15490:
15486:
15480:
15463:
15451:
15439:
15424:
15391:
15382:
15373:
15360:
14997:
14851:
14724:
13590:
13587:
13573:
13421:
13418:
13397:
13285:
13089:
13038:
12769:
12685:
12564:and a point
12543:
12534:open mapping
12136:
12052:
11874:
11580:
11408:
11228:
10991:
10829:
10568:
10566:
9972:
9858:
9693:of the form
9559:
9343:
9201:
9131:
9083:
9077:
8945:
8732:
8707:
8685:
8567:
8566:
7769:
7275:
7270:
7266:
7258:
6912:with centre
5954:
5947:as required.
5416:choose some
4665:
4654:
4511:
4459:
4432:Definitions.
4431:
4430:
4385:vanishes at
4076:
4051:
3513:and then to
3194:
2601:with length
2386:
2043:
1956:
1841:
1778:
1680:
1321:
1320:
1222:contractible
993:
965:
937:
923:
890:
888:in general.
848:Lars Ahlfors
817:
805:
779:is equal to
683:
591:
524:
519:
517:
313:
307:
128:Basic theory
27:
16818:Riemann sum
16460:: 310–314,
16063:: 107–144,
15884:(6): 2577.
15816:Nehari 1952
15793:Schiff 1993
15770:Schiff 1993
15713:Schiff 1993
15668:Jänich 1993
15634:Jänich 1993
15612:Jänich 1993
15537:Conway 1978
15446:, p. 1
15417:Osgood 1900
15397:Osgood 1900
14786:as long as
14017:as long as
13579:quasicircle
13571:curve or a
12617:which maps
8790:containing
7455:tending to
6990:can lie in
6941:and radius
4629:. A family
1231:(e.g., the
1198:(as can be
948:Lipót Fejér
422:holomorphic
344:open subset
242:Kiyoshi Oka
61:Real number
16943:Categories
16587:0387979670
16482:31.0420.01
16375:Birkhäuser
16326:3540563377
16287:0005.06802
16253:1604.04071
16044:0387973494
16004:0471996718
15983:0070006571
15963:References
15921:(2): 221.
15759:Duren 1983
15748:Duren 1980
15702:Duren 1980
15657:Duren 1983
15623:Duren 1983
15601:Duren 1983
15401:Walsh 1973
13415:Algorithms
12282:. Suppose
11490:, so that
10830:such that
8521:, so that
7233:, so that
5111:of radius
4890:open with
4691:is normal.
990:Koch curve
962:Importance
944:Paul Koebe
16675:EMS Press
16634:0002-9890
16474:0002-9947
16281:: 15–36,
16077:115544426
15886:CiteSeerX
15852:Bell 1992
15456:Gray 1994
15429:Gray 1994
15405:Gray 1994
15041:Ω
15018:Ω
14967:Ω
14864:Ω
14822:−
14799:ϕ
14736:ϕ
14596:→
14577:Ω
14568:ϕ
14483:Ω
14480:∂
14422:Ω
14419:∈
14389:Ω
14331:and time
14294:⋅
14222:and time
14207:≥
14157:Ω
14154:∂
14117:−
14092:Ω
14089:∂
14061:−
14053:−
14030:ϕ
14000:−
13966:ϕ
13940:Ω
13937:∂
13898:Ω
13834:and time
13787:−
13747:→
13728:Ω
13719:ϕ
13689:×
13659:Ω
13656:∂
13636:Ω
13633:∈
13603:Ω
13528:γ
13525:∈
13509:…
13476:γ
13443:…
13365:unit disk
13363:, or the
13009:−
12998:
12992:−
12951:∂
12948:∈
12720:−
12582:∈
12189:−
12035:⋯
12016:−
11988:−
11961:−
11950:−
11857:⋯
11838:−
11810:−
11783:−
11772:−
11748:∘
11697:∈
11391:⋯
11380:−
11324:−
11211:⋯
11200:−
11125:≤
11037:≤
10992:and thus
10974:⋯
10963:−
10812:⋯
10801:−
10732:⊃
10699:∞
10693:∪
10552:⋯
10387:∞
10297:∞
10291:∖
10242:≤
10197:≤
10108:≤
10001:∂
9957:⋯
9946:−
9884:∞
9878:∪
9870:⊂
9733:⋯
9722:−
9540:−
9529:−
9423:≤
9405:−
9329:⋯
9318:−
9243:≤
9187:⋯
9088:in 1909.
9063:θ
8999:θ
8990:−
8954:∞
8931:⋯
8881:−
8798:∞
8775:∞
8769:∪
8718:θ
8579:ϕ
8436:−
8339:−
8262:∈
8237:θ
8231:−
8130:¯
8111:−
8088:−
8068:θ
7923:¯
7915:−
7895:−
7828:∖
7802:≠
7709:∈
7397:∈
7360:∈
7303:∞
7300:≤
7259:non-empty
7194:∘
7058:Δ
7055:−
7052:∖
6978:Δ
6975:−
6900:Δ
6848:∈
6786:−
6783:≠
6719:−
6676:∖
6668:∈
6426:α
6389:¯
6386:α
6381:−
6364:α
6361:−
6350:θ
6218:−
6210:∘
5996:∈
5968:≠
5875:δ
5866:δ
5857:≤
5830:−
5762:−
5694:−
5667:≤
5640:−
5584:∈
5561:δ
5381:δ
5375:≤
5364:−
5350:≤
5323:−
5256:′
5232:∫
5206:−
5119:δ
5066:δ
5023:≤
4901:⊂
4434:A family
4361:−
4295:−
4131:−
4117:∫
4107:π
4027:⊂
4021:∂
4018:∪
3952:−
3909:∂
3726:−
3612:δ
3609:−
3576:≥
3553:δ
3544:−
3474:δ
3471:−
3278:based at
3260:δ
3233:∈
3203:γ
3154:γ
3087:γ
2993:−
2963:∂
2959:∫
2952:π
2888:. Taking
2873:∈
2864:γ
2780:∂
2760:∅
2751:∩
2700:δ
2697:≤
2609:δ
2521:δ
2478:∈
2475:∞
2412:∖
2406:∞
2400:∪
2285:γ
2245:γ
2178:γ
2175:∘
2102:−
2052:γ
1983:
1902:γ
1855:−
1787:γ
1692:∈
1662:∞
1656:∪
1541:∉
1418:vanishes;
1335:⊂
1282:≥
1225:manifolds
787:ϕ
670:ϕ
470:∈
444:onto the
419:bijective
338:non-empty
16927:Category
16559:citation
16544:(1851),
16515:(1998),
16499:(1931),
16421:(1952),
16348:(2007),
16208:(1994),
16178:(2001),
16157:(1983),
16111:(1995),
16086:(1978),
15993:(1979),
15972:(1978),
15953:14545404
15330:See also
15282:for any
13870:, where
13403:or from
12107:of each
10156:lies in
9477:and set
8466:′
8381:′
8315:′
8295:′
8205:′
8178:′
7852:and let
7669:′
7609:′
7433:′
7327:′
7279:, after
7146:′
6863:. Since
6629:′
6282:′
6118:′
5177:. Since
5014:′
4976:′
4541:lies in
4246:and set
4152:′
4007:. Hence
3821:lies in
3250:homotopy
2344:for all
2137:outside
1530:for any
1322:Theorem.
1299:polydisk
979:boundary
820:boundary
730:argument
417:(i.e. a
397:mapping
16650:0323996
16642:2318448
16490:1986285
16225:1295591
15933:Bibcode
13314:, then
11292:. Then
10399:of the
10212:, then
9380:, then
9112:due to
9081:-axis.
9075:to the
8730:to the
8568:Remark.
8251:. Then
7379:. Pick
7281:Ahlfors
4870:. Take
4512:compact
3125:equals
2935:, then
2740:. Then
950:and to
886:domains
814:History
346:of the
16648:
16640:
16632:
16584:
16531:
16488:
16480:
16472:
16433:
16381:
16356:
16323:
16301:
16285:
16223:
16186:
16165:
16144:
16123:
16098:
16075:
16041:
16019:
16001:
15980:
15951:
15888:
15201:Then,
13338:, the
12770:where
12544:Given
12532:is an
10176:with
10052:. For
8166:where
8032:. Let
7749:takes
7283:. Let
7209:is in
7019:. Let
5470:where
4460:normal
3901:is on
3624:. Let
3172:about
3105:about
2297:about
2257:about
2190:about
2021:where
1894:along
1556:, the
1088:annuli
312:, the
213:People
16638:JSTOR
16551:(PDF)
16486:JSTOR
16248:arXiv
16213:(PDF)
16073:S2CID
15949:S2CID
15923:arXiv
15472:(PDF)
15385:: 3–4
15353:Notes
13488:with
13290:: If
11265:onto
10571:that
9973:take
9202:with
8946:near
8850:with
8658:onto
8012:into
7637:. So
7416:with
7263:Koebe
7070:onto
6582:with
6449:. So
6413:with
4405:, so
3967:, so
3881:; if
3841:. If
3038:over
2720:from
2467:with
1914:from
1799:from
1209:(see
1054:with
336:is a
16630:ISSN
16582:ISBN
16565:link
16529:ISBN
16470:ISSN
16431:ISBN
16379:ISBN
16354:ISBN
16321:ISBN
16299:ISBN
16184:ISBN
16163:ISBN
16142:ISBN
16121:ISBN
16096:ISBN
16039:ISBN
16017:ISBN
15999:ISBN
15978:ISBN
15523:See
15299:<
15293:<
15246:CONF
15162:CONF
14993:#SAT
14884:>
14816:<
14242:<
14047:<
13910:and
13238:and
12937:for
11585:and
11522:>
11441:<
11250:>
10063:>
10037:<
9841:>
9678:>
9560:for
9462:>
9365:>
9104:and
9096:and
8459:>
7770:onto
7697:and
7682:>
7622:>
7597:and
7562:has
7349:for
7294:<
7168:>
7134:and
6952:>
6817:for
6642:>
6617:and
6434:<
6295:>
6270:and
6173:and
6131:>
6106:and
5533:and
5122:>
5069:>
4698:Let
3781:off
3718:and
3565:for
3380:<
3263:>
2772:and
2559:and
2524:>
2493:and
2447:and
2117:for
1229:ball
1180:<
1164:<
1126:<
1110:<
1071:<
1065:<
1036:<
1020:<
662:and
494:<
16622:doi
16478:JFM
16462:doi
16408:doi
16283:Zbl
16258:doi
16065:doi
15941:doi
15896:doi
15495:doi
15234:AC0
15232:is
15212:MAJ
15152:by
14524:by
14302:max
12995:log
12664:to
12418:in
12080:or
10866:is
10680:in
10660:of
9830:in
8830:on
7952:on
7817:in
7273:of
7257:is
6736:in
6542:of
6031:of
5957:If
5443:in
5038:on
4752:of
4510:is
3944:is
3691:to
3486:to
2815:or
2649:of
2210:is
1980:exp
1934:to
1819:to
1600:is
1560:of
1507:on
1464:on
935:).
822:of
308:In
16945::
16673:,
16667:,
16646:MR
16644:,
16636:,
16628:,
16618:80
16616:,
16580:,
16561:}}
16557:{{
16527:,
16523:,
16503:,
16484:,
16476:,
16468:,
16452:,
16429:,
16425:,
16404:20
16402:,
16373:,
16319:,
16279:84
16256:,
16246:,
16242:,
16234:;
16221:MR
16215:,
16119:,
16115:,
16094:,
16090:,
16071:,
16061:72
16059:,
16055:,
16037:,
15947:.
15939:.
15931:.
15919:45
15917:.
15894:.
15882:45
15880:.
15491:20
15489:.
15003:#P
14935:20
13648:.
13585:.
13411:.
13390:.
13087:.
12536:.
9856:.
9116:.
8678:.
5606:,
3801:,
2384:.
1776:.
1235:).
1213:).
1202:).
803:.
522:.
26:→
16704:e
16697:t
16690:v
16624::
16567:)
16464::
16454:1
16410::
16260::
16250::
16244:3
16067::
15955:.
15943::
15935::
15925::
15902:.
15898::
15866:.
15501:.
15497::
15474:.
15368:.
15314:.
15308:2
15305:1
15296:c
15290:0
15270:)
15265:c
15261:n
15257:,
15254:n
15251:(
15218:n
15189:.
15186:)
15181:c
15177:n
15173:,
15170:n
15167:(
15138:c
15134:n
15129:/
15125:1
15105:)
15100:2
15096:n
15092:(
15089:O
15069:n
15065:/
15061:1
15021:,
14998:n
14995:(
14979:,
14976:)
14973:0
14970:,
14964:(
14961:r
14938:n
14915:n
14895:2
14891:/
14887:1
14837:.
14831:n
14828:1
14819:1
14812:|
14808:)
14805:w
14802:(
14795:|
14771:n
14768:1
14745:)
14742:w
14739:(
14725:n
14708:k
14704:2
14700:=
14697:n
14677:)
14674:k
14671:(
14668:O
14648:)
14645:n
14641:/
14637:1
14634:(
14631:O
14611:)
14608:0
14605:,
14602:D
14599:(
14593:)
14588:0
14584:w
14580:,
14574:(
14571::
14548:)
14543:2
14539:n
14535:(
14532:O
14509:n
14506:1
14461:,
14456:k
14452:2
14448:=
14445:n
14425:.
14414:0
14410:w
14365:.
14360:)
14355:a
14351:n
14347:(
14344:O
14340:2
14317:)
14314:2
14311:,
14308:a
14305:(
14298:n
14291:C
14271:,
14266:)
14261:a
14257:n
14253:(
14250:O
14246:2
14239:)
14236:n
14233:(
14230:T
14210:1
14204:a
14182:a
14178:n
14134:.
14129:)
14126:n
14123:(
14120:O
14113:2
14069:.
14064:n
14057:2
14050:1
14043:|
14039:)
14036:w
14033:(
14026:|
14003:n
13996:2
13975:)
13972:w
13969:(
13946:.
13943:)
13934:,
13929:0
13925:w
13921:(
13918:d
13878:C
13856:)
13853:n
13850:(
13847:O
13843:2
13820:2
13816:n
13812:C
13790:n
13783:2
13762:)
13759:0
13756:,
13753:D
13750:(
13744:)
13739:0
13735:w
13731:,
13725:(
13722::
13697:n
13693:2
13684:n
13680:2
13628:0
13624:w
13577:-
13574:K
13557:1
13553:C
13531:.
13520:n
13516:z
13512:,
13506:,
13501:0
13497:z
13454:n
13450:z
13446:,
13440:,
13435:0
13431:z
13374:D
13350:C
13322:U
13298:U
13266:f
13246:v
13226:u
13206:U
13186:v
13166:g
13142:u
13118:U
13098:u
13067:u
13047:u
13023:|
13017:0
13013:z
13006:z
13002:|
12989:=
12986:)
12983:z
12980:(
12977:u
12954:U
12945:z
12925:1
12922:=
12918:|
12914:)
12911:z
12908:(
12905:f
12901:|
12880:f
12858:0
12854:z
12833:v
12813:u
12793:v
12790:i
12787:+
12784:u
12781:=
12778:g
12755:,
12750:)
12747:z
12744:(
12741:g
12737:e
12733:)
12728:0
12724:z
12717:z
12714:(
12711:=
12708:)
12705:z
12702:(
12699:f
12686:U
12672:0
12650:0
12646:z
12625:U
12605:f
12585:U
12577:0
12573:z
12552:U
12518:2
12514:F
12491:i
12487:C
12466:G
12446:t
12426:G
12406:t
12403:=
12400:)
12397:w
12394:(
12389:2
12385:F
12360:t
12340:)
12335:i
12331:C
12327:(
12322:2
12318:F
12295:2
12291:F
12270:0
12267:=
12264:)
12261:a
12258:(
12253:2
12249:F
12228:G
12208:)
12205:w
12202:(
12197:1
12193:F
12186:)
12183:w
12180:(
12175:0
12171:F
12167:=
12164:)
12161:w
12158:(
12153:2
12149:F
12137:y
12120:i
12116:C
12093:1
12089:F
12066:0
12062:F
12038:.
12032:+
12027:2
12023:)
12019:a
12013:w
12010:(
12005:2
12001:b
11997:+
11994:)
11991:a
11985:w
11982:(
11977:1
11973:b
11969:+
11964:1
11957:)
11953:a
11947:w
11944:(
11941:=
11938:)
11935:w
11932:(
11927:1
11923:F
11899:)
11896:w
11893:(
11888:1
11884:F
11860:,
11854:+
11849:2
11845:)
11841:a
11835:w
11832:(
11827:2
11823:a
11819:+
11816:)
11813:a
11807:w
11804:(
11799:1
11795:a
11791:+
11786:1
11779:)
11775:a
11769:w
11766:(
11763:=
11760:)
11757:w
11754:(
11751:f
11745:h
11742:=
11739:)
11736:w
11733:(
11728:0
11724:F
11700:G
11694:a
11672:i
11668:C
11645:0
11641:C
11620:G
11600:h
11566:)
11561:1
11557:b
11553:(
11549:e
11546:R
11525:0
11519:)
11514:1
11510:b
11506:(
11502:e
11499:R
11478:)
11473:1
11469:c
11465:+
11460:1
11456:b
11452:(
11448:e
11445:R
11438:)
11433:1
11429:c
11425:(
11421:e
11418:R
11394:.
11388:+
11383:1
11376:w
11372:)
11367:1
11363:c
11359:+
11354:1
11350:a
11346:(
11343:+
11340:w
11337:=
11332:0
11328:c
11321:)
11318:)
11315:w
11312:(
11309:k
11306:(
11303:f
11278:2
11274:G
11253:S
11246:|
11242:w
11238:|
11214:,
11208:+
11203:1
11196:w
11190:1
11186:c
11182:+
11177:0
11173:c
11169:+
11166:w
11163:=
11160:)
11157:w
11154:(
11151:k
11128:0
11122:)
11117:1
11113:b
11109:(
11105:e
11102:R
11081:f
11061:)
11056:1
11052:a
11048:(
11044:e
11041:R
11034:)
11029:1
11025:b
11021:+
11016:1
11012:a
11008:(
11004:e
11001:R
10977:,
10971:+
10966:1
10959:z
10955:)
10950:1
10946:b
10942:+
10937:1
10933:a
10929:(
10926:+
10923:z
10920:=
10917:)
10914:)
10911:z
10908:(
10905:f
10902:(
10899:h
10875:C
10854:)
10849:2
10845:G
10841:(
10838:h
10815:,
10809:+
10804:1
10797:w
10791:1
10787:b
10783:+
10780:w
10777:=
10774:)
10771:w
10768:(
10765:h
10740:1
10736:G
10727:2
10723:G
10702:}
10696:{
10689:C
10668:K
10646:2
10642:G
10621:K
10599:1
10595:G
10591:=
10588:)
10585:G
10582:(
10579:f
10549:+
10544:1
10540:a
10536:+
10533:z
10530:=
10527:)
10524:z
10521:(
10518:f
10495:)
10490:1
10486:a
10482:(
10478:e
10475:R
10454:f
10434:f
10412:n
10408:f
10367:f
10347:f
10325:n
10321:f
10300:}
10294:{
10288:G
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9922:=
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7119:=
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6711:=
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6636:a
6633:(
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6593:(
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6570:D
6550:G
6530:f
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6495:.
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6480:=
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6339:=
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6330:(
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6094:0
6091:=
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4964:{
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4924:E
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4346:=
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4340:z
4337:(
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4314:)
4311:a
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4299:f
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4264:(
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4225:f
4222:=
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4216:a
4213:(
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3652:0
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3539:0
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3531:=
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3522:w
3499:0
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3466:0
3462:z
3439:0
3435:z
3408:1
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3383:N
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3333:N
3313:N
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2996:a
2990:z
2987:(
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2945:1
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2876:G
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2823:B
2803:A
2783:C
2757:=
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2748:C
2728:A
2708:4
2704:/
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2657:A
2637:a
2617:4
2613:/
2588:C
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2527:0
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2415:G
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2403:{
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2099:z
2077:n
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2029:f
2009:)
2006:2
2002:/
1998:)
1995:x
1992:(
1989:f
1986:(
1977:=
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1971:z
1968:(
1965:g
1942:x
1922:a
1882:z
1878:d
1873:/
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1865:d
1858:1
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1744:z
1740:d
1735:f
1715:a
1695:G
1689:a
1665:}
1659:{
1652:C
1631:G
1620:;
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1406:G
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1285:2
1279:n
1271:(
1257:n
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1186:}
1183:4
1176:|
1172:z
1168:|
1161:1
1158::
1155:z
1152:{
1132:}
1129:2
1122:|
1118:z
1114:|
1107:1
1104::
1101:z
1098:{
1074:1
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1062:0
1042:}
1039:1
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1028:z
1024:|
1017:r
1014::
1011:z
1008:{
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908:C
872:U
830:U
765:0
761:z
740:f
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713:=
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705:0
701:z
697:(
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684:f
650:U
628:0
624:z
603:f
573:f
553:U
533:U
503:.
500:}
497:1
490:|
486:z
482:|
478::
474:C
467:z
464:{
461:=
458:D
432:U
405:f
380:C
358:C
324:U
297:e
290:t
283:v
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