2064:
113:
Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation
2484:
1745:
There is no corresponding notion of ramification for transcendental and logarithmic branch points since the associated covering
Riemann surface cannot be analytically continued to a cover of the branch point itself. Such covers are therefore always unramified.
215:
is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial
2801:
1960:
is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always, taken between pairs of branch points.
2222:
One reason that branch cuts are common features of complex analysis is that a branch cut can be thought of as a sum of infinitely many poles arranged along a line in the complex plane with infinitesimal residues. For example,
1947:
Roughly speaking, branch points are the points where the various sheets of a multiple valued function come together. The branches of the function are the various sheets of the function. For example, the function
232:
typically means the former more restrictive kind: the algebraic branch points. In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.
2497:) with a cut from −1 to 1. The branch cut can be moved around, since the integration line can be shifted without altering the value of the integral so long as the line does not pass across the point
2213:
i. These surfaces are glued to each other along the branch cut in the unique way to make the logarithm continuous. Each time the variable goes around the origin, the logarithm moves to a different branch.
968:
2039:
The branch cut device may appear arbitrary (and it is); but it is very useful, for example in the theory of special functions. An invariant explanation of the branch phenomenon is developed in
1657:
2300:
2018:
1023:. However, the inverse of a holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a
1964:
Branch cuts allow one to work with a collection of single-valued functions, "glued" together along the branch cut instead of a multivalued function. For example, to make the function
322:
2157:
2281:
3011:
1677:
If the monodromy group is infinite, that is, it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about
1308:
1526:
1377:
1078:
1476:
2023:
single-valued, one makes a branch cut along the interval on the real axis, connecting the two branch points of the function. The same idea can be applied to the function
1737:
288:
1447:
1215:
635:
602:
517:
404:
1403:
3066:
795:
671:
2628:
1341:
1248:
560:
428:
259:
145:
1173:
1128:
994:
873:
846:
826:
732:
705:
455:
1098:
1017:
756:
537:
475:
370:
346:
213:
185:
165:
104:
84:
64:
2686:
2190:
in a connected open set in the complex plane. In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a
1130:. More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, such as those that are defined
1799:
will have made one half-circle, going from the positive square root of 4, i.e., from 2, to the negative square root of 4, i.e., −2.
1699:
at the origin. Going once counterclockwise around a simple closed curve encircling the origin, the complex logarithm is incremented by 2
2540:
where ƒ fails to be a cover are the ramification points of ƒ, and the image of a ramification point under ƒ is called a branch point.
2209:, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2
2891:). The degree of ƒ is defined to be the degree of this field extension , and ƒ is said to be finite if the degree is finite.
3287:
3260:
3238:
3216:
3127:
2067:
A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number
2479:{\displaystyle u(z)=\int _{a=-1}^{a=1}f_{a}(z)\,da=\int _{a=-1}^{a=1}{1 \over z-a}\,da=\log \left({z+1 \over z-1}\right)}
878:
1594:
1970:
191:
of any solution around a closed loop containing the origin will result in a different function: there is non-trivial
2194:. A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience.
2664:, then there is no need to select special coordinates. The ramification index can be calculated explicitly from
759:
293:
2036:
is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis.
3339:
2111:
2229:
2961:
2665:
1956:
has two branches: one where the square root comes in with a plus sign, and the other with a minus sign. A
349:
3334:
3175:
1175:, branch points are those points around which there is nontrivial monodromy. For example, the function
3360:
3208:
1253:
675:
17:
3355:
3274:
2071:
goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a
1938:
for irrational α has a logarithmic branch point, and its derivative is singular without being a pole.
1481:
225:
1346:
1034:
2641:. Usually the ramification index is one. But if the ramification index is not equal to one, then
1452:
1024:
3329:
1720:
271:
1911:
1896:
1139:
1408:
1178:
611:
565:
480:
1382:
3045:
1695:. This is so called because the typical example of this phenomenon is the branch point of the
764:
640:
3316:, Translated and edited by Richard A. Silverman, Englewood Cliffs, N.J.: Prentice-Hall Inc.,
2600:
2052:
1575:
1567:
1541:
1313:
1220:
545:
413:
244:
221:
188:
117:
2043:
theory (of which it is historically the origin), and more generally in the ramification and
1148:
1103:
973:
3321:
3297:
1028:
851:
831:
804:
710:
683:
433:
375:
325:
43:
8:
2842:
2201:
i when crossing the branch cut. The logarithm can be made continuous by gluing together
1588:
An example of a transcendental branch point is the origin for the multi-valued function
2848:
2796:{\displaystyle e_{P}={\frac {1}{2\pi i}}\int _{\gamma }{\frac {f'(z)}{f(z)-f(P)}}\,dz.}
2174:
will yield another possible angle. A branch of the logarithm is a continuous function
2048:
1083:
1002:
741:
522:
460:
355:
331:
198:
170:
150:
89:
69:
49:
1134:. A unifying framework for dealing with such examples is supplied in the language of
3301:
3283:
3256:
3234:
3212:
3123:
2836:
2564:
2080:
1803:
1696:
1131:
3269:
3115:
2084:
1020:
86:
values) at that point, all of its neighborhoods contain a point that has more than
35:
3317:
3293:
3279:
3107:
2924:
2880:
2852:
2518:
2040:
1578:
of a function element once around some simple closed curve surrounding the point
1135:
107:
3119:
3252:
3026:
2851:, the notion of branch points can be generalized to mappings between arbitrary
2533:
2529:
2090:
605:
1670:
group for a circuit around the origin is finite. Analytic continuation around
3349:
2202:
1780:
1742:
Logarithmic branch points are special cases of transcendental branch points.
266:
3305:
1844:
2863:
be a morphism of algebraic curves. By pulling back rational functions on
3230:
1756:
407:
31:
2509:
The concept of a branch point is defined for a holomorphic function ƒ:
2063:
2044:
1667:
217:
192:
3114:, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 213–269,
2536:
onto its image at all but a finite number of points. The points of
2089:
The typical example of a branch cut is the complex logarithm. If a
1863:/4 are among the multiple values of arctan(1). The imaginary units
1142:
of order greater than 1 can also be considered ramification points.
1899:
at those two points, since the denominator is zero at those points.
262:
110:, and the formal definition of branch points employs this concept.
2806:
This integral is the number of times ƒ(γ) winds around the point
1528:. Thus there is monodromy around this loop enclosing the origin.
1776:
1772:
1531:
828:
is a ramification point if there exists a holomorphic function
2947:) = 0 whose differential is nonzero. Pulling back
2162:
However, there is an obvious ambiguity in defining the angle
106:
values. Multi-valued functions are rigorously studied using
1879:)log. This may be seen by observing that the derivative (
3207:, Cambridge Texts in Applied Mathematics (2nd ed.),
3176:"Logarithmic branch point - Encyclopedia of Mathematics"
1795:
has made one full circle, going from 4 back to 4 again,
1674:
full circuits brings the function back to the original.
3048:
2964:
2689:
2603:
2303:
2232:
2114:
1973:
1723:
1717:
and the monodromy group is the infinite cyclic group
1597:
1484:
1455:
1411:
1385:
1349:
1316:
1256:
1223:
1181:
1151:
1138:
below. In particular, in this more general picture,
1106:
1086:
1037:
1005:
976:
881:
854:
834:
807:
767:
744:
713:
707:. If the ramification index is greater than 1, then
686:
643:
614:
568:
548:
525:
483:
463:
436:
416:
378:
358:
334:
296:
274:
247:
201:
173:
153:
120:
92:
72:
52:
3081: > 1, then ƒ is said to be ramified at
2637:. This integer is called the ramification index of
2532:). Unless it is constant, the function ƒ will be a
1871:
are branch points of the arctangent function arctan(
195:. Despite the algebraic branch point, the function
187:. Here the branch point is the origin, because the
27:
Point of interest for complex multi-valued functions
2935:is a regular function defined in a neighborhood of
963:{\displaystyle f(z)=\phi (z)(z-z_{0})^{k}+f(z_{0})}
3060:
3005:
2795:
2622:
2478:
2275:
2151:
2012:
1731:
1651:
1520:
1470:
1441:
1397:
1371:
1335:
1302:
1242:
1209:
1167:
1122:
1100:as a branch point of the global analytic function
1092:
1072:
1011:
988:
962:
867:
840:
820:
789:
750:
726:
699:
665:
629:
596:
554:
531:
511:
469:
449:
422:
398:
364:
340:
316:
282:
253:
207:
179:
159:
139:
98:
78:
58:
3314:Theory of functions of a complex variable. Vol. I
3203:Ablowitz, Mark J.; Fokas, Athanassios S. (2003),
1652:{\displaystyle g(z)=\exp \left(z^{-1/k}\right)\,}
1145:In terms of the inverse global analytic function
3347:
3205:Complex Variables: Introduction and Applications
3108:"Fractional Differintegrations Insight Concepts"
2032:; but in that case one has to perceive that the
1926:. The converse is not true, since the function
1825:moves along a circle of radius 1 centered at 0,
2013:{\displaystyle F(z)={\sqrt {z}}{\sqrt {1-z}}\,}
2294:. Integrating over the location of the pole:
3202:
3311:
3163:
1821:are among the multiple values of ln(1). As
1532:Transcendental and logarithmic branch points
604:, taken with its positive orientation. The
3327:
3246:
3159:
2645:is by definition a ramification point, and
2197:The logarithm has a jump discontinuity of 2
1540:is a global analytic function defined on a
1250:. The inverse function is the square root
3268:
3029:in the local ring of regular functions at
236:
2783:
2668:. Let γ be a simple rectifiable loop in
2426:
2368:
2148:
2009:
1855:/4) are both equal to 1, the two numbers
1725:
1706:. Encircling a loop with winding number
1648:
1585:produces a different function element.
317:{\displaystyle f:\Omega \to \mathbb {C} }
310:
276:
2062:
1343:. Indeed, going around the closed loop
46:is a point such that if the function is
3224:
3148:
2152:{\displaystyle \ln z=\ln r+i\theta .\,}
1783:centered at 0. The dependent variable
372:, that is, the zeros of the derivative
14:
3348:
2894:Assume that ƒ is finite. For a point
2276:{\displaystyle f_{a}(z)={1 \over z-a}}
1449:. But after going around the loop to
519:containing no other critical point of
3006:{\displaystyle e_{P}=v_{P}(t\circ f)}
2830:
2217:
348:is not constant, then the set of the
3247:Arfken, G. B.; Weber, H. J. (2000),
2286:is a function with a simple pole at
2058:
999:Typically, one is not interested in
3249:Mathematical Methods for Physicists
3105:
2951:by ƒ defines a regular function on
2504:
1710:, the logarithm is incremented by 2
24:
2676:. The ramification index of ƒ at
1922:has a logarithmic branch point at
417:
303:
248:
25:
3372:
2590:in terms of which the function ƒ(
1303:{\displaystyle f^{-1}(w)=w^{1/2}}
673:is a positive integer called the
2656:is just the Riemann sphere, and
1802:0 is also a branch point of the
1521:{\displaystyle e^{2\pi i/2}=-1}
3168:
3153:
3142:
3112:Functional Fractional Calculus
3099:
3000:
2988:
2777:
2771:
2762:
2756:
2748:
2742:
2365:
2359:
2313:
2307:
2249:
2243:
1983:
1977:
1942:
1791:in a continuous manner. When
1771:starts at 4 and moves along a
1607:
1601:
1372:{\displaystyle w=e^{i\theta }}
1310:, which has a branch point at
1276:
1270:
1191:
1185:
1073:{\displaystyle w_{0}=f(z_{0})}
1067:
1054:
957:
944:
929:
909:
906:
900:
891:
885:
784:
771:
660:
647:
624:
618:
591:
572:
506:
487:
393:
387:
306:
228:, unqualified use of the term
13:
1:
3196:
2524:to a compact Riemann surface
2093:is represented in polar form
1666: > 1. Here the
1471:{\displaystyle \theta =2\pi }
848:defined in a neighborhood of
477:lies at the center of a disc
3312:Markushevich, A. I. (1965),
3251:(5th ed.), Boston, MA:
2925:local uniformizing parameter
2911:is defined as follows. Let
2814:is a ramification point and
1732:{\displaystyle \mathbb {Z} }
1217:has a ramification point at
1031:and refer to a branch point
283:{\displaystyle \mathbb {C} }
7:
3335:Encyclopedia of Mathematics
3328:Solomentsev, E.D. (2001) ,
3120:10.1007/978-3-642-20545-3_5
1787:changes while depending on
1755:0 is a branch point of the
1749:
1557:transcendental branch point
10:
3377:
3209:Cambridge University Press
3180:www.encyclopediaofmath.org
3089:is called a branch point.
2840:
2834:
2078:
1442:{\displaystyle e^{i0/2}=1}
1210:{\displaystyle f(z)=z^{2}}
637:with respect to the point
630:{\displaystyle f(\gamma )}
597:{\displaystyle B(z_{0},r)}
512:{\displaystyle B(z_{0},r)}
430:. So each critical point
2902:, the ramification index
2867:to rational functions on
2666:Cauchy's integral formula
2660:is in the finite part of
2517:from a compact connected
2170:any integer multiple of 2
2101:e, then the logarithm of
1398:{\displaystyle \theta =0}
797:is called an (algebraic)
226:geometric function theory
3092:
3061:{\displaystyle t\circ f}
2563:, there are holomorphic
2182:) giving a logarithm of
1693:logarithmic branch point
1025:global analytic function
790:{\displaystyle f(z_{0})}
758:, and the corresponding
666:{\displaystyle f(z_{0})}
3225:Ahlfors, L. V. (1979),
2623:{\displaystyle w=z^{k}}
1336:{\displaystyle w_{0}=0}
1243:{\displaystyle z_{0}=0}
555:{\displaystyle \gamma }
423:{\displaystyle \Omega }
254:{\displaystyle \Omega }
237:Algebraic branch points
140:{\displaystyle w^{2}=z}
3106:Das, Shantanu (2011),
3062:
3042:is the order to which
3007:
2797:
2624:
2480:
2277:
2153:
2076:
2053:differential equations
2014:
1906: ' of a function
1733:
1653:
1522:
1472:
1443:
1399:
1373:
1337:
1304:
1244:
1211:
1169:
1168:{\displaystyle f^{-1}}
1124:
1123:{\displaystyle f^{-1}}
1094:
1074:
1013:
990:
989:{\displaystyle k>1}
964:
869:
842:
822:
791:
752:
728:
701:
667:
631:
598:
556:
533:
513:
471:
451:
424:
400:
366:
342:
318:
284:
255:
209:
181:
161:
141:
100:
80:
60:
3063:
3008:
2818:is a branch point if
2798:
2625:
2481:
2278:
2154:
2066:
2015:
1891:) = 1/(1 +
1734:
1654:
1576:analytic continuation
1568:essential singularity
1523:
1473:
1444:
1400:
1374:
1338:
1305:
1245:
1212:
1170:
1125:
1095:
1075:
1014:
991:
965:
870:
868:{\displaystyle z_{0}}
843:
841:{\displaystyle \phi }
823:
821:{\displaystyle z_{0}}
792:
753:
729:
727:{\displaystyle z_{0}}
702:
700:{\displaystyle z_{0}}
668:
632:
599:
557:
534:
514:
472:
452:
450:{\displaystyle z_{0}}
425:
401:
399:{\displaystyle f'(z)}
367:
343:
319:
285:
256:
222:essential singularity
210:
189:analytic continuation
182:
162:
142:
101:
81:
61:
3278:, Berlin, New York:
3046:
2962:
2687:
2601:
2301:
2230:
2205:many copies, called
2112:
1971:
1721:
1595:
1482:
1453:
1409:
1383:
1347:
1314:
1254:
1221:
1179:
1149:
1104:
1084:
1035:
1003:
974:
879:
852:
832:
805:
765:
742:
711:
684:
641:
612:
566:
546:
523:
481:
461:
434:
414:
376:
356:
332:
326:holomorphic function
294:
272:
245:
199:
171:
151:
118:
90:
70:
50:
44:multivalued function
2843:Unramified morphism
2827: > 1.
2649:is a branch point.
2489:defines a function
2407:
2348:
2049:algebraic functions
1759:function. Suppose
1027:. It is common to
1019:itself, but in its
562:be the boundary of
3275:Algebraic Geometry
3058:
3003:
2849:algebraic geometry
2847:In the context of
2831:Algebraic geometry
2793:
2620:
2476:
2378:
2319:
2273:
2218:Continuum of poles
2149:
2077:
2010:
1902:If the derivative
1833:) goes from 0 to 2
1729:
1649:
1518:
1468:
1439:
1395:
1369:
1333:
1300:
1240:
1207:
1165:
1120:
1090:
1070:
1009:
986:
960:
865:
838:
818:
787:
748:
736:ramification point
724:
697:
663:
627:
594:
552:
529:
509:
467:
447:
420:
396:
362:
338:
314:
280:
251:
205:
177:
157:
137:
96:
76:
56:
3361:Inverse functions
3289:978-0-387-90244-9
3270:Hartshorne, Robin
3262:978-0-12-059825-0
3240:978-0-07-000657-7
3218:978-0-521-53429-1
3164:Markushevich 1965
3129:978-3-642-20544-6
3085:. In that case,
2837:Branched covering
2781:
2719:
2633:for some integer
2565:local coordinates
2470:
2424:
2271:
2081:Complex logarithm
2059:Complex logarithm
2034:point at infinity
2007:
1994:
1804:natural logarithm
1697:complex logarithm
1684:, then the point
1662:for some integer
1379:, one starts at
1093:{\displaystyle f}
1012:{\displaystyle f}
801:. Equivalently,
751:{\displaystyle f}
532:{\displaystyle f}
470:{\displaystyle f}
365:{\displaystyle f}
341:{\displaystyle f}
208:{\displaystyle w}
180:{\displaystyle z}
167:as a function of
160:{\displaystyle w}
99:{\displaystyle n}
79:{\displaystyle n}
59:{\displaystyle n}
16:(Redirected from
3368:
3356:Complex analysis
3342:
3324:
3308:
3265:
3243:
3227:Complex Analysis
3221:
3190:
3189:
3187:
3186:
3172:
3166:
3160:Solomentsev 2001
3157:
3151:
3146:
3140:
3138:
3137:
3136:
3103:
3067:
3065:
3064:
3059:
3012:
3010:
3009:
3004:
2987:
2986:
2974:
2973:
2853:algebraic curves
2802:
2800:
2799:
2794:
2782:
2780:
2751:
2741:
2732:
2730:
2729:
2720:
2718:
2704:
2699:
2698:
2629:
2627:
2626:
2621:
2619:
2618:
2505:Riemann surfaces
2485:
2483:
2482:
2477:
2475:
2471:
2469:
2458:
2447:
2425:
2423:
2409:
2406:
2395:
2358:
2357:
2347:
2336:
2282:
2280:
2279:
2274:
2272:
2270:
2256:
2242:
2241:
2212:
2200:
2173:
2158:
2156:
2155:
2150:
2085:Principal branch
2075:of the function.
2031:
2030:
2019:
2017:
2016:
2011:
2008:
1997:
1995:
1990:
1862:
1858:
1854:
1850:
1836:
1817:
1738:
1736:
1735:
1730:
1728:
1713:
1702:
1658:
1656:
1655:
1650:
1647:
1643:
1642:
1638:
1527:
1525:
1524:
1519:
1508:
1507:
1503:
1477:
1475:
1474:
1469:
1448:
1446:
1445:
1440:
1432:
1431:
1427:
1404:
1402:
1401:
1396:
1378:
1376:
1375:
1370:
1368:
1367:
1342:
1340:
1339:
1334:
1326:
1325:
1309:
1307:
1306:
1301:
1299:
1298:
1294:
1269:
1268:
1249:
1247:
1246:
1241:
1233:
1232:
1216:
1214:
1213:
1208:
1206:
1205:
1174:
1172:
1171:
1166:
1164:
1163:
1136:Riemann surfaces
1129:
1127:
1126:
1121:
1119:
1118:
1099:
1097:
1096:
1091:
1079:
1077:
1076:
1071:
1066:
1065:
1047:
1046:
1021:inverse function
1018:
1016:
1015:
1010:
995:
993:
992:
987:
969:
967:
966:
961:
956:
955:
937:
936:
927:
926:
874:
872:
871:
866:
864:
863:
847:
845:
844:
839:
827:
825:
824:
819:
817:
816:
796:
794:
793:
788:
783:
782:
757:
755:
754:
749:
733:
731:
730:
725:
723:
722:
706:
704:
703:
698:
696:
695:
672:
670:
669:
664:
659:
658:
636:
634:
633:
628:
603:
601:
600:
595:
584:
583:
561:
559:
558:
553:
539:in its closure.
538:
536:
535:
530:
518:
516:
515:
510:
499:
498:
476:
474:
473:
468:
456:
454:
453:
448:
446:
445:
429:
427:
426:
421:
405:
403:
402:
397:
386:
371:
369:
368:
363:
347:
345:
344:
339:
323:
321:
320:
315:
313:
289:
287:
286:
281:
279:
260:
258:
257:
252:
214:
212:
211:
206:
186:
184:
183:
178:
166:
164:
163:
158:
146:
144:
143:
138:
130:
129:
108:Riemann surfaces
105:
103:
102:
97:
85:
83:
82:
77:
65:
63:
62:
57:
36:complex analysis
21:
3376:
3375:
3371:
3370:
3369:
3367:
3366:
3365:
3346:
3345:
3290:
3280:Springer-Verlag
3263:
3241:
3219:
3199:
3194:
3193:
3184:
3182:
3174:
3173:
3169:
3158:
3154:
3147:
3143:
3134:
3132:
3130:
3104:
3100:
3095:
3080:
3047:
3044:
3043:
3041:
3024:
2982:
2978:
2969:
2965:
2963:
2960:
2959:
2915: = ƒ(
2910:
2881:field extension
2845:
2839:
2833:
2826:
2752:
2734:
2733:
2731:
2725:
2721:
2708:
2703:
2694:
2690:
2688:
2685:
2684:
2614:
2610:
2602:
2599:
2598:
2555: = ƒ(
2519:Riemann surface
2507:
2459:
2448:
2446:
2442:
2413:
2408:
2396:
2382:
2353:
2349:
2337:
2323:
2302:
2299:
2298:
2260:
2255:
2237:
2233:
2231:
2228:
2227:
2220:
2210:
2198:
2171:
2113:
2110:
2109:
2087:
2079:Main articles:
2061:
2041:Riemann surface
2026:
2024:
1996:
1989:
1972:
1969:
1968:
1945:
1860:
1856:
1852:
1848:
1834:
1815:
1810:is the same as
1752:
1724:
1722:
1719:
1718:
1711:
1700:
1690:
1683:
1634:
1627:
1623:
1619:
1596:
1593:
1592:
1584:
1565:
1550:
1534:
1499:
1489:
1485:
1483:
1480:
1479:
1454:
1451:
1450:
1423:
1416:
1412:
1410:
1407:
1406:
1384:
1381:
1380:
1360:
1356:
1348:
1345:
1344:
1321:
1317:
1315:
1312:
1311:
1290:
1286:
1282:
1261:
1257:
1255:
1252:
1251:
1228:
1224:
1222:
1219:
1218:
1201:
1197:
1180:
1177:
1176:
1156:
1152:
1150:
1147:
1146:
1111:
1107:
1105:
1102:
1101:
1085:
1082:
1081:
1061:
1057:
1042:
1038:
1036:
1033:
1032:
1004:
1001:
1000:
975:
972:
971:
951:
947:
932:
928:
922:
918:
880:
877:
876:
859:
855:
853:
850:
849:
833:
830:
829:
812:
808:
806:
803:
802:
778:
774:
766:
763:
762:
743:
740:
739:
718:
714:
712:
709:
708:
691:
687:
685:
682:
681:
654:
650:
642:
639:
638:
613:
610:
609:
579:
575:
567:
564:
563:
547:
544:
543:
524:
521:
520:
494:
490:
482:
479:
478:
462:
459:
458:
441:
437:
435:
432:
431:
415:
412:
411:
379:
377:
374:
373:
357:
354:
353:
350:critical points
333:
330:
329:
309:
295:
292:
291:
275:
273:
270:
269:
261:be a connected
246:
243:
242:
239:
200:
197:
196:
172:
169:
168:
152:
149:
148:
125:
121:
119:
116:
115:
91:
88:
87:
71:
68:
67:
51:
48:
47:
28:
23:
22:
15:
12:
11:
5:
3374:
3364:
3363:
3358:
3344:
3343:
3330:"Branch point"
3325:
3309:
3288:
3266:
3261:
3253:Academic Press
3244:
3239:
3222:
3217:
3198:
3195:
3192:
3191:
3167:
3152:
3141:
3128:
3097:
3096:
3094:
3091:
3076:
3057:
3054:
3051:
3037:
3020:
3014:
3013:
3002:
2999:
2996:
2993:
2990:
2985:
2981:
2977:
2972:
2968:
2906:
2835:Main article:
2832:
2829:
2822:
2804:
2803:
2792:
2789:
2786:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2750:
2747:
2744:
2740:
2737:
2728:
2724:
2717:
2714:
2711:
2707:
2702:
2697:
2693:
2631:
2630:
2617:
2613:
2609:
2606:
2594:) is given by
2559:) ∈
2543:For any point
2530:Riemann sphere
2506:
2503:
2487:
2486:
2474:
2468:
2465:
2462:
2457:
2454:
2451:
2445:
2441:
2438:
2435:
2432:
2429:
2422:
2419:
2416:
2412:
2405:
2402:
2399:
2394:
2391:
2388:
2385:
2381:
2377:
2374:
2371:
2367:
2364:
2361:
2356:
2352:
2346:
2343:
2340:
2335:
2332:
2329:
2326:
2322:
2318:
2315:
2312:
2309:
2306:
2284:
2283:
2269:
2266:
2263:
2259:
2254:
2251:
2248:
2245:
2240:
2236:
2219:
2216:
2160:
2159:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2091:complex number
2060:
2057:
2021:
2020:
2006:
2003:
2000:
1993:
1988:
1985:
1982:
1979:
1976:
1944:
1941:
1940:
1939:
1900:
1851:/4) and tan (5
1841:
1814:, both 0 and 2
1800:
1751:
1748:
1727:
1688:
1681:
1660:
1659:
1646:
1641:
1637:
1633:
1630:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1582:
1563:
1548:
1542:punctured disc
1533:
1530:
1517:
1514:
1511:
1506:
1502:
1498:
1495:
1492:
1488:
1467:
1464:
1461:
1458:
1438:
1435:
1430:
1426:
1422:
1419:
1415:
1394:
1391:
1388:
1366:
1363:
1359:
1355:
1352:
1332:
1329:
1324:
1320:
1297:
1293:
1289:
1285:
1281:
1278:
1275:
1272:
1267:
1264:
1260:
1239:
1236:
1231:
1227:
1204:
1200:
1196:
1193:
1190:
1187:
1184:
1162:
1159:
1155:
1117:
1114:
1110:
1089:
1069:
1064:
1060:
1056:
1053:
1050:
1045:
1041:
1029:abuse language
1008:
985:
982:
979:
959:
954:
950:
946:
943:
940:
935:
931:
925:
921:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
862:
858:
837:
815:
811:
786:
781:
777:
773:
770:
760:critical value
747:
721:
717:
694:
690:
662:
657:
653:
649:
646:
626:
623:
620:
617:
606:winding number
593:
590:
587:
582:
578:
574:
571:
551:
528:
508:
505:
502:
497:
493:
489:
486:
466:
444:
440:
419:
395:
392:
389:
385:
382:
361:
337:
312:
308:
305:
302:
299:
278:
250:
238:
235:
204:
176:
156:
136:
133:
128:
124:
95:
75:
55:
26:
9:
6:
4:
3:
2:
3373:
3362:
3359:
3357:
3354:
3353:
3351:
3341:
3337:
3336:
3331:
3326:
3323:
3319:
3315:
3310:
3307:
3303:
3299:
3295:
3291:
3285:
3281:
3277:
3276:
3271:
3267:
3264:
3258:
3254:
3250:
3245:
3242:
3236:
3232:
3228:
3223:
3220:
3214:
3210:
3206:
3201:
3200:
3181:
3177:
3171:
3165:
3161:
3156:
3150:
3145:
3131:
3125:
3121:
3117:
3113:
3109:
3102:
3098:
3090:
3088:
3084:
3079:
3075:
3071:
3055:
3052:
3049:
3040:
3036:
3032:
3028:
3023:
3019:
2997:
2994:
2991:
2983:
2979:
2975:
2970:
2966:
2958:
2957:
2956:
2954:
2950:
2946:
2942:
2938:
2934:
2930:
2926:
2922:
2918:
2914:
2909:
2905:
2901:
2898: ∈
2897:
2892:
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2859: →
2858:
2854:
2850:
2844:
2838:
2828:
2825:
2821:
2817:
2813:
2810:. As above,
2809:
2790:
2787:
2784:
2774:
2768:
2765:
2759:
2753:
2745:
2738:
2735:
2726:
2722:
2715:
2712:
2709:
2705:
2700:
2695:
2691:
2683:
2682:
2681:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2650:
2648:
2644:
2640:
2636:
2615:
2611:
2607:
2604:
2597:
2596:
2595:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2566:
2562:
2558:
2554:
2550:
2547: ∈
2546:
2541:
2539:
2535:
2531:
2528:(usually the
2527:
2523:
2520:
2516:
2513: →
2512:
2502:
2500:
2496:
2492:
2472:
2466:
2463:
2460:
2455:
2452:
2449:
2443:
2439:
2436:
2433:
2430:
2427:
2420:
2417:
2414:
2410:
2403:
2400:
2397:
2392:
2389:
2386:
2383:
2379:
2375:
2372:
2369:
2362:
2354:
2350:
2344:
2341:
2338:
2333:
2330:
2327:
2324:
2320:
2316:
2310:
2304:
2297:
2296:
2295:
2293:
2290: =
2289:
2267:
2264:
2261:
2257:
2252:
2246:
2238:
2234:
2226:
2225:
2224:
2215:
2208:
2204:
2195:
2193:
2189:
2185:
2181:
2177:
2169:
2165:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2108:
2107:
2106:
2104:
2100:
2097: =
2096:
2092:
2086:
2082:
2074:
2070:
2065:
2056:
2054:
2050:
2046:
2042:
2037:
2035:
2029:
2004:
2001:
1998:
1991:
1986:
1980:
1974:
1967:
1966:
1965:
1962:
1959:
1955:
1952: =
1951:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1910:has a simple
1909:
1905:
1901:
1898:
1895:) has simple
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1846:
1842:
1839:
1832:
1828:
1824:
1820:
1813:
1809:
1805:
1801:
1798:
1794:
1790:
1786:
1782:
1781:complex plane
1778:
1774:
1770:
1766:
1763: =
1762:
1758:
1754:
1753:
1747:
1743:
1740:
1716:
1709:
1705:
1698:
1694:
1687:
1680:
1675:
1673:
1669:
1665:
1644:
1639:
1635:
1631:
1628:
1624:
1620:
1616:
1613:
1610:
1604:
1598:
1591:
1590:
1589:
1586:
1581:
1577:
1573:
1569:
1562:
1558:
1554:
1547:
1543:
1539:
1536:Suppose that
1529:
1515:
1512:
1509:
1504:
1500:
1496:
1493:
1490:
1486:
1465:
1462:
1459:
1456:
1436:
1433:
1428:
1424:
1420:
1417:
1413:
1392:
1389:
1386:
1364:
1361:
1357:
1353:
1350:
1330:
1327:
1322:
1318:
1295:
1291:
1287:
1283:
1279:
1273:
1265:
1262:
1258:
1237:
1234:
1229:
1225:
1202:
1198:
1194:
1188:
1182:
1160:
1157:
1153:
1143:
1141:
1137:
1133:
1115:
1112:
1108:
1087:
1062:
1058:
1051:
1048:
1043:
1039:
1030:
1026:
1022:
1006:
997:
983:
980:
977:
952:
948:
941:
938:
933:
923:
919:
915:
912:
903:
897:
894:
888:
882:
860:
856:
835:
813:
809:
800:
779:
775:
768:
761:
745:
737:
719:
715:
692:
688:
679:
677:
655:
651:
644:
621:
615:
607:
588:
585:
580:
576:
569:
549:
540:
526:
503:
500:
495:
491:
484:
464:
442:
438:
409:
390:
383:
380:
359:
351:
335:
327:
300:
297:
268:
267:complex plane
264:
234:
231:
227:
223:
219:
202:
194:
190:
174:
154:
134:
131:
126:
122:
111:
109:
93:
73:
66:-valued (has
53:
45:
41:
37:
33:
19:
3333:
3313:
3273:
3248:
3229:, New York:
3226:
3204:
3183:. Retrieved
3179:
3170:
3155:
3149:Ahlfors 1979
3144:
3133:, retrieved
3111:
3101:
3086:
3082:
3077:
3073:
3069:
3068:vanishes at
3038:
3034:
3033:. That is,
3030:
3021:
3017:
3015:
2952:
2948:
2944:
2940:
2936:
2932:
2928:
2920:
2916:
2912:
2907:
2903:
2899:
2895:
2893:
2888:
2884:
2876:
2872:
2868:
2864:
2860:
2856:
2846:
2823:
2819:
2815:
2811:
2807:
2805:
2677:
2673:
2669:
2661:
2657:
2653:
2651:
2646:
2642:
2638:
2634:
2632:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2560:
2556:
2552:
2548:
2544:
2542:
2537:
2534:covering map
2525:
2521:
2514:
2510:
2508:
2498:
2494:
2490:
2488:
2291:
2287:
2285:
2221:
2206:
2196:
2191:
2187:
2183:
2179:
2175:
2167:
2166:: adding to
2163:
2161:
2102:
2098:
2094:
2088:
2073:branch point
2072:
2068:
2038:
2033:
2027:
2022:
1963:
1957:
1953:
1949:
1946:
1935:
1931:
1927:
1923:
1919:
1915:
1907:
1903:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1847:, since tan(
1845:trigonometry
1837:
1830:
1826:
1822:
1818:
1811:
1807:
1796:
1792:
1788:
1784:
1768:
1764:
1760:
1744:
1741:
1714:
1707:
1703:
1692:
1691:is called a
1685:
1678:
1676:
1671:
1663:
1661:
1587:
1579:
1571:
1560:
1556:
1552:
1545:
1537:
1535:
1144:
998:
970:for integer
799:branch point
798:
735:
734:is called a
676:ramification
674:
541:
240:
230:branch point
229:
112:
40:branch point
39:
32:mathematical
29:
3231:McGraw-Hill
2931:; that is,
1943:Branch cuts
1914:at a point
1867:and −
1757:square root
408:limit point
3350:Categories
3197:References
3185:2019-06-11
3135:2022-04-27
2919:) and let
2841:See also:
2192:branch cut
2047:theory of
1958:branch cut
1574:such that
1478:, one has
1132:implicitly
875:such that
18:Branch cut
3340:EMS Press
3053:∘
3027:valuation
2995:∘
2855:. Let ƒ:
2766:−
2727:γ
2723:∫
2713:π
2464:−
2440:
2418:−
2390:−
2380:∫
2331:−
2321:∫
2265:−
2203:countably
2143:θ
2131:
2119:
2045:monodromy
2002:−
1887:) arctan(
1806:. Since
1779:4 in the
1668:monodromy
1629:−
1617:
1513:−
1494:π
1466:π
1457:θ
1387:θ
1365:θ
1263:−
1158:−
1113:−
916:−
898:ϕ
836:ϕ
622:γ
550:γ
418:Ω
406:, has no
307:→
304:Ω
249:Ω
218:monodromy
193:monodromy
34:field of
3306:13348052
3272:(1977),
3139:(page 6)
2955:. Then
2739:′
2186:for all
1875:) = (1/2
1859:/4 and 5
1750:Examples
1715:i w
1551:. Then
384:′
263:open set
3322:0171899
3298:0463157
3025:is the
2879:) is a
2672:around
2025:√
1918:, then
1544:around
265:in the
220:and an
30:In the
3320:
3304:
3296:
3286:
3259:
3237:
3215:
3126:
3072:. If
3016:where
2207:sheets
1777:radius
1773:circle
1767:, and
1566:is an
1555:has a
328:. If
224:. In
3093:Notes
2939:with
2923:be a
2586:near
2574:near
1897:poles
1829:= ln(
1140:poles
678:index
42:of a
3302:OCLC
3284:ISBN
3257:ISBN
3235:ISBN
3213:ISBN
3124:ISBN
2582:for
2578:and
2570:for
2551:and
2083:and
2051:and
1934:) =
1912:pole
1405:and
981:>
542:Let
290:and
241:Let
147:for
38:, a
3116:doi
2927:at
2883:of
2680:is
2652:If
2437:log
2105:is
1843:In
1775:of
1614:exp
1570:of
1559:if
1080:of
738:of
680:of
608:of
457:of
410:in
352:of
3352::
3338:,
3332:,
3318:MR
3300:,
3294:MR
3292:,
3282:,
3255:,
3233:,
3211:,
3178:.
3162:;
3122:,
3110:,
2871:,
2501:.
2128:ln
2116:ln
2055:.
1908:ƒ
1885:dz
1739:.
996:.
324:a
3188:.
3118::
3087:Q
3083:P
3078:P
3074:e
3070:P
3056:f
3050:t
3039:P
3035:e
3031:P
3022:P
3018:v
3001:)
2998:f
2992:t
2989:(
2984:P
2980:v
2976:=
2971:P
2967:e
2953:X
2949:t
2945:Q
2943:(
2941:t
2937:Q
2933:t
2929:P
2921:t
2917:P
2913:Q
2908:P
2904:e
2900:X
2896:P
2889:Y
2887:(
2885:K
2877:X
2875:(
2873:K
2869:X
2865:Y
2861:Y
2857:X
2824:P
2820:e
2816:Q
2812:P
2808:Q
2791:.
2788:z
2785:d
2778:)
2775:P
2772:(
2769:f
2763:)
2760:z
2757:(
2754:f
2749:)
2746:z
2743:(
2736:f
2716:i
2710:2
2706:1
2701:=
2696:P
2692:e
2678:P
2674:P
2670:X
2662:Y
2658:Q
2654:Y
2647:Q
2643:P
2639:P
2635:k
2616:k
2612:z
2608:=
2605:w
2592:z
2588:Q
2584:Y
2580:w
2576:P
2572:X
2568:z
2561:Y
2557:P
2553:Q
2549:X
2545:P
2538:X
2526:Y
2522:X
2515:Y
2511:X
2499:z
2495:z
2493:(
2491:u
2473:)
2467:1
2461:z
2456:1
2453:+
2450:z
2444:(
2434:=
2431:a
2428:d
2421:a
2415:z
2411:1
2404:1
2401:=
2398:a
2393:1
2387:=
2384:a
2376:=
2373:a
2370:d
2366:)
2363:z
2360:(
2355:a
2351:f
2345:1
2342:=
2339:a
2334:1
2328:=
2325:a
2317:=
2314:)
2311:z
2308:(
2305:u
2292:a
2288:z
2268:a
2262:z
2258:1
2253:=
2250:)
2247:z
2244:(
2239:a
2235:f
2211:π
2199:π
2188:z
2184:z
2180:z
2178:(
2176:L
2172:π
2168:θ
2164:θ
2146:.
2140:i
2137:+
2134:r
2125:=
2122:z
2103:z
2099:r
2095:z
2069:z
2028:z
2005:z
1999:1
1992:z
1987:=
1984:)
1981:z
1978:(
1975:F
1954:z
1950:w
1936:z
1932:z
1930:(
1928:ƒ
1924:a
1920:ƒ
1916:a
1904:ƒ
1893:z
1889:z
1883:/
1881:d
1877:i
1873:z
1869:i
1865:i
1861:π
1857:π
1853:π
1849:π
1840:.
1838:i
1835:π
1831:z
1827:w
1823:z
1819:i
1816:π
1812:e
1808:e
1797:w
1793:z
1789:z
1785:w
1769:z
1765:z
1761:w
1726:Z
1712:π
1708:w
1704:i
1701:π
1689:0
1686:z
1682:0
1679:z
1672:k
1664:k
1645:)
1640:k
1636:/
1632:1
1625:z
1621:(
1611:=
1608:)
1605:z
1602:(
1599:g
1583:0
1580:z
1572:g
1564:0
1561:z
1553:g
1549:0
1546:z
1538:g
1516:1
1510:=
1505:2
1501:/
1497:i
1491:2
1487:e
1463:2
1460:=
1437:1
1434:=
1429:2
1425:/
1421:0
1418:i
1414:e
1393:0
1390:=
1362:i
1358:e
1354:=
1351:w
1331:0
1328:=
1323:0
1319:w
1296:2
1292:/
1288:1
1284:w
1280:=
1277:)
1274:w
1271:(
1266:1
1259:f
1238:0
1235:=
1230:0
1226:z
1203:2
1199:z
1195:=
1192:)
1189:z
1186:(
1183:f
1161:1
1154:f
1116:1
1109:f
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1068:)
1063:0
1059:z
1055:(
1052:f
1049:=
1044:0
1040:w
1007:f
984:1
978:k
958:)
953:0
949:z
945:(
942:f
939:+
934:k
930:)
924:0
920:z
913:z
910:(
907:)
904:z
901:(
895:=
892:)
889:z
886:(
883:f
861:0
857:z
814:0
810:z
785:)
780:0
776:z
772:(
769:f
746:f
720:0
716:z
693:0
689:z
661:)
656:0
652:z
648:(
645:f
625:)
619:(
616:f
592:)
589:r
586:,
581:0
577:z
573:(
570:B
527:f
507:)
504:r
501:,
496:0
492:z
488:(
485:B
465:f
443:0
439:z
394:)
391:z
388:(
381:f
360:f
336:f
311:C
301::
298:f
277:C
203:w
175:z
155:w
135:z
132:=
127:2
123:w
94:n
74:n
54:n
20:)
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