7342:
3502:
6115:. In that case, actual "steepest descent contours" are defined and computed. The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on the external field, was done in
3875:
3232:
4966:
6126:, especially convenient when jump matrices do not have analytic extensions. Their method is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours. An alternative way of dealing with jump matrices with no analytic extensions was introduced in
6039:
In particular, Riemann–Hilbert factorization problems are used to extract asymptotic values for the three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity).
5263:
4808:
3277:
3645:
1650:
was simple. A full
Riemann–Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The "+" and "−" sides of the "contour" may then be determined according to the index of a point with respect to
4192:
5687:
4384:
5995:
above, it is one of the original rigorous investigations of so-called "integrable probability". But the connection between the theory of integrability and various classical ensembles of random matrices goes back to the work of Dyson (see e.g.
5424:
3067:
4819:
1204:
5106:
3607:
4710:
821:
1298:
3497:{\displaystyle \log M={\frac {1}{2\pi i}}\int _{\Sigma }{\frac {\log {2}}{\zeta -z}}d\zeta ={\frac {\log 2}{2\pi i}}\int _{-1-z}^{1-z}{\frac {1}{\zeta }}d\zeta ={\frac {\log 2}{2\pi i}}\log {\frac {z-1}{z+1}},}
6051:
By analogy with the classical asymptotic methods, one "deforms" Riemann–Hilbert problems which are not explicitly solvable to problems that are. The so-called "nonlinear" method of stationary phase is due to
1893:
1530:
6099:
has in fact provided the basis for the analysis of most of the work concerning "real" orthogonal polynomials (i.e. with the orthogonality condition defined on the real line) and
Hermitian random matrices.
4519:
5539:
6981:
McLaughlin, K.; Miller, P. (2006), "The d-bar steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights",
3982:
3870:{\displaystyle {\begin{aligned}M_{+}(0)&=(e^{i\pi })^{\frac {\log 2}{2\pi i}}=e^{\frac {\log 2}{2}}\\M_{-}(0)&=(e^{-i\pi })^{\frac {\log 2}{2\pi i}}=e^{-{\frac {\log 2}{2}}}\end{aligned}}}
688:
3650:
2822:
6149:. The related singular kernel is not the usual Cauchy kernel, but rather a more general kernel involving meromorphic differentials defined naturally on the surface (see e.g. the appendix in
968:
4197:
is also a solution. In general, conditions on growth are necessary at special points (the end-points of the jump contour or intersection point) to ensure that the problem is well-posed.
6160:
Most
Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by
4059:
4419:
416:
2288:
6103:
Perhaps the most sophisticated extension of the theory so far is the one applied to the "non self-adjoint" case, i.e. when the underlying Lax operator (the first component of the
250:
5570:
852:
5295:
5845:
5801:
5737:
5098:
895:
3227:{\displaystyle {\frac {1}{2\pi i}}\int _{\Sigma _{+}}{\frac {\log 2}{\zeta -z}}\,d\zeta -{\frac {1}{2\pi i}}\int _{\Sigma _{-}}{\frac {\log {2}}{\zeta -z}}\,d\zeta =\log 2}
2985:
5070:
4273:
3261:
2932:
2737:
2395:
1790:
1430:
445:
5307:
2906:
280:
171:
144:
5894:
1983:
1925:
1620:
1562:
6087:, which has been crucial in most applications. This was inspired by work of Lax, Levermore and Venakides, who reduced the analysis of the small dispersion limit of the
5972:). Furthermore, the distribution of eigenvalues of random matrices in several classical ensembles is reduced to computations involving orthogonal polynomials (see e.g.
4961:{\displaystyle {\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=g(x,y),\quad {\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=h(x,y).}
1954:
1591:
2647:
1744:
1708:
1384:
1348:
1043:
1007:
724:
3634:
3059:
2552:
2415:
2308:
2218:
2120:
2067:
2011:
1764:
1669:
1648:
1404:
621:
590:
570:
327:
85:
5044:
5015:
4662:
4627:
4592:
4557:
3039:
3012:
2530:
2502:
2475:
2369:
2178:
2151:
307:
5461:
2854:
2448:
2339:
2100:
2047:
532:
503:
474:
2606:
4051:
5865:
5757:
5710:
5562:
5481:
4989:
4702:
4682:
4263:
4234:
4028:
4008:
2874:
2687:
2667:
2572:
2198:
115:
6742:
Its, A.R. (1982), "Asymptotics of
Solutions of the Nonlinear Schrödinger Equation and Isomonodromic Deformations of Systems of Linear Differential Equations",
5258:{\displaystyle M(z,{\bar {z}})=1+{\frac {1}{2\pi i}}\iint _{\mathbb {R} ^{2}}{\frac {f(\zeta ,{\bar {\zeta }})}{\zeta -z}}\,d\zeta \wedge d{\bar {\zeta }},}
1055:
6083:
An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by
3513:
4803:{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}
6091:
to the analysis of a maximization problem for a logarithmic potential under some external field: a variational problem of "electrostatic" type (see
6072:. A crucial ingredient of the Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard
5968:
Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a
Riemann–Hilbert factorization problem (
6887:
Kuijlaars, Arno; López, Abey (2015), "A vector equilibrium problem for the normal matrix model, and multiple orthogonal polynomials on a star",
6939:
6400:
735:
7302:
6119:; the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov.
1215:
6375:
6015:
studies a class of
Riemann-Hilbert problems coming from Donaldson-Thomas theory and makes connections with Gromov-Witten theory and exact
6422:
177:
of the contour with respect to a point. The classical problem, considered in
Riemann's PhD dissertation, was that of finding a function
7122:
6095:). The g-function is the logarithmic transform of the maximizing "equilibrium" measure. The analysis of the small dispersion limit of
4424:
1798:
7157:
3264:
1438:
4433:
7403:
5489:
16:
For the original problem of
Hilbert concerning the existence of linear differential equations having a given monodromy group, see
7408:
7162:
5904:
6665:
6626:
6587:
3886:
7060:
7001:
6506:
6359:
6529:; Zhou, X. (1993), "A Steepest Descent Method for Oscillatory Riemann–Hilbert Problems; Asymptotics for the MKdV Equation",
5991:
on the distribution of the length of the longest increasing subsequence of a random permutation. Together with the study of
629:
2745:
6787:
Kamvissis, S.; Rakhmanov, E.A. (2005), "Existence and
Regularity for an Energy Maximization Problem in Two Dimensions",
6040:
There exists a method for extracting the asymptotic behavior of solutions of
Riemann–Hilbert problems, analogous to the
7277:
7267:
7252:
7172:
6177:
3987:
CAVEAT 1: If the problem is not scalar one cannot easily take logarithms. In general explicit solutions are very rare.
2013:(technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane), a
903:
7312:
17:
7373:
7327:
6023:
The numerical analysis of Riemann–Hilbert problems can provide an effective way for numerically solving integrable
6153:). The Riemann–Hilbert problem deformation theory is applied to the problem of stability of the infinite periodic
7023:
6789:
6663:
Fokas, A.S.; Its, A.R.; Kitaev, A.V. (1992), "The isomonodromy approach to matrix models in 2D quantum gravity",
4187:{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}}+{\frac {a}{z-1}}+{\frac {b}{z+1}}}
7413:
7322:
7317:
7292:
7147:
7115:
6518:
New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann–Hilbert Problems
5938:
4240:
6734:
6716:
6706:
6498:
6432:
6024:
4397:
3268:
335:
6724:
7398:
7257:
7142:
5908:
2223:
727:
5682:{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=A(z,{\bar {z}})M+B(z,{\bar {z}}){\overline {M}},}
183:
7378:
6729:
6711:
6427:
7393:
7388:
7307:
7287:
6045:
6041:
829:
7072:
Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions
6370:
5271:
7418:
7368:
7346:
7108:
5806:
5762:
5715:
5075:
4379:{\displaystyle {\frac {\partial M(z,{\bar {z}})}{\partial {\bar {z}}}}=f(z,{\bar {z}}),\quad z\in D,}
5419:{\displaystyle d\zeta \wedge d{\bar {\zeta }}=(d\xi +id\eta )\wedge (d\xi -id\eta )=-2id\xi d\eta .}
3263:. Because the solution of a Riemann–Hilbert factorization problem is unique (an easy application of
857:
7297:
7187:
7152:
6441:
6146:
2944:
5049:
3240:
2911:
2699:
2374:
1769:
1409:
424:
7207:
6157:
under a "short range" perturbation (for example a perturbation of a finite number of particles).
6006:
2879:
600:
258:
149:
122:
6371:"On the distribution of the length of the longest increasing subsequence of random permutations"
5870:
1959:
1901:
1596:
1538:
7423:
7222:
5959:
4237:
1930:
1567:
6840:(2012), "Long-time asymptotics of the periodic Toda lattice under short-range perturbations",
6398:
Beals, R.; Coifman, R.R. (1984), "Scattering and inverse scattering for first order systems",
2611:
1713:
1677:
1353:
1317:
1012:
976:
693:
7247:
6994:
Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations
6531:
5900:
3619:
3044:
2537:
2400:
2293:
2203:
2105:
2052:
1996:
1749:
1654:
1633:
1389:
606:
575:
537:
312:
70:
40:
6122:
An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in
5950:
5023:
4994:
4632:
4597:
4562:
4527:
2450:
is smooth and integrable. In more complicated cases it could have singularities. The limits
7032:
6969:
6906:
6859:
6808:
6674:
6635:
6596:
6460:
6142:
3017:
2990:
2508:
2480:
2453:
2347:
2156:
2129:
1672:
285:
92:
6484:, Oper. Theory: Advances and Appl., vol. 3, Basel-Boston-Stuttgart: Birkhäuser Verlag
5437:
2830:
2424:
2315:
2076:
2023:
508:
479:
450:
8:
7272:
7182:
7167:
2585:
7036:
6973:
6910:
6863:
6812:
6678:
6639:
6600:
6464:
4033:
2574:
near those points have to be posed to ensure uniqueness (see the scalar problem below).
7192:
6922:
6896:
6875:
6849:
6824:
6798:
6690:
6651:
6612:
6566:
6558:
6540:
6450:
6182:
5930:
5850:
5742:
5695:
5547:
5466:
4974:
4687:
4667:
4248:
4219:
4013:
3993:
3613:
2859:
2672:
2652:
2557:
2183:
1314:
Hilbert's generalization of the problem attempted to find a pair of analytic functions
1309:
100:
7383:
7262:
7202:
7056:
6997:
6926:
6918:
6694:
6655:
6616:
6502:
6355:
6137:
where the underlying space of the Riemann–Hilbert problem is a compact hyperelliptic
6016:
2938:
2505:
1199:{\displaystyle {\frac {a(z)+ib(z)}{2}}M_{+}(z)+{\frac {a(z)-ib(z)}{2}}M_{-}(z)=c(z),}
48:
6828:
6570:
7197:
7177:
7131:
7089:
7040:
6948:
6914:
6879:
6867:
6816:
6781:, Annals of Mathematics Study, vol. 154, Princeton: Princeton University Press
6682:
6643:
6604:
6550:
6468:
6409:
6384:
5924:
5919:
Riemann–Hilbert problems have applications to several related classes of problems.
32:
6937:; Levermore, C.D. (1983), "The Zero Dispersion Limit for the KdV Equation I-III",
6389:
5941:
on the line, or to periodic problems, or even to initial-boundary value problems (
7242:
7217:
6757:
6624:
Fokas, A.S. (2002), "Integrable nonlinear evolution equations on the half-line",
6582:
6138:
5963:
2342:
6439:
Bridgeland, T. (2019), "Riemann–Hilbert problems from Donaldson–Thomas theory",
3602:{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}},}
7232:
7227:
6779:
Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation
6161:
5934:
593:
174:
56:
7021:
Varzugin, G.G. (1996), "Asymptotics of oscillatory Riemann-Hilbert problems",
6647:
6472:
7362:
6837:
6702:
6578:
6073:
5298:
4244:
96:
44:
36:
7212:
6952:
6934:
6413:
6154:
6108:
6096:
6088:
4211:
624:
7237:
6526:
6490:
5982:
2070:
816:{\displaystyle M_{-}(z)={\overline {M_{+}\left({\bar {z}}^{-1}\right)}}.}
24:
7080:
Zhou, Xin (1989), "The Riemann–Hilbert problem and inverse scattering",
2577:
1630:
In the Riemann problem as well as Hilbert's generalization, the contour
1622:
are given complex-valued functions (no longer just complex conjugates).
534:
are given real-valued functions. For example, in the special case where
6686:
6608:
6562:
6141:. The correct factorization problem is no more holomorphic, but rather
3990:
CAVEAT 2: The boundedness (or at least a constraint on the blow-up) of
88:
52:
6871:
6820:
1293:{\displaystyle \lim _{z\to \infty }M_{-}(z)={\overline {{M}_{+}(0)}}.}
7044:
6545:
5946:
2693:
1046:
7093:
6554:
5945:), can be stated as a Riemann–Hilbert problem. Likewise the inverse
5564:. For generalized analytic functions, this equation is replaced by
7100:
6455:
6104:
6901:
6854:
6803:
2554:, the jump condition is not defined; constraints on the growth of
973:
Hence the problem reduces to finding a pair of analytic functions
6482:
Factorization of matrix functions and singular integral operators
7013:
The Hilbert transform of Schwartz distributions and applications
2504:
could be classical and continuous or they could be taken in the
1888:{\displaystyle \alpha (t)M_{+}(t)+\beta (z)M_{-}(t)=\gamma (t).}
6520:, International Mathematical Research Notices, pp. 286–299
1525:{\displaystyle \alpha (t)M_{+}(t)+\beta (t)M_{-}(t)=\gamma (t)}
4514:{\displaystyle M=u+iv,\quad f={\frac {g+ih}{2}},\quad z=x+iy,}
6207:
4389:
is a generalization of a Riemann-Hilbert problem, called the
6327:
6303:
6279:
5534:{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=0,}
3041:
are the limits of the Cauchy transform from above and below
6960:
Manakov, S.V. (1974), "Nonlinear Fraunnhofer diffraction",
6291:
6243:
6776:
6195:
6112:
3977:{\displaystyle M_{+}(0)=M_{-}(0)e^{\log {2}}=M_{-}(0)2.}
1209:
and, moreover, so that the condition at infinity holds:
6583:"Fredholm Determinants and Inverse Scattering Problems"
5914:
5017:, then the Cauchy-Riemann equations must be satisfied.
2122:, such that the following two conditions are satisfied
2534:. At end-points or intersection points of the contour
1386:
on the inside and outside, respectively, of the curve
632:
6423:"Boundary value problems of analytic function theory"
6255:
5873:
5853:
5809:
5765:
5745:
5718:
5698:
5573:
5550:
5492:
5469:
5440:
5310:
5274:
5268:
integrated over the entire complex plane; denoted by
5109:
5078:
5052:
5026:
4997:
4977:
4822:
4713:
4690:
4670:
4635:
4600:
4565:
4530:
4436:
4400:
4276:
4251:
4222:
4062:
4036:
4016:
3996:
3889:
3648:
3622:
3516:
3280:
3243:
3070:
3047:
3020:
2993:
2947:
2914:
2882:
2862:
2833:
2748:
2702:
2675:
2655:
2614:
2588:
2578:
Example: Scalar Riemann–Hilbert factorization problem
2560:
2540:
2511:
2483:
2456:
2427:
2403:
2377:
2350:
2318:
2296:
2226:
2206:
2186:
2159:
2132:
2108:
2079:
2055:
2026:
1999:
1962:
1933:
1904:
1801:
1772:
1752:
1716:
1680:
1657:
1636:
1599:
1570:
1541:
1441:
1412:
1392:
1356:
1320:
1218:
1058:
1015:
979:
906:
860:
832:
738:
696:
683:{\textstyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}
609:
578:
540:
511:
482:
453:
427:
338:
315:
288:
261:
186:
152:
125:
103:
73:
39:, are a class of problems that arise in the study of
6758:"The Riemann–Hilbert Problem and Integrable Systems"
6267:
5899:
Generalized analytic functions have applications in
1671:. The Riemann–Hilbert problem is to find a pair of
6219:
2817:{\displaystyle \log M_{+}(z)=\log M_{-}(z)+\log 2.}
51:for Riemann–Hilbert problems have been produced by
6786:
6777:Kamvissis, S.; McLaughlin, K.; Miller, P. (2003),
6515:
6368:
6116:
6084:
5988:
5888:
5859:
5839:
5795:
5751:
5731:
5704:
5681:
5556:
5533:
5475:
5455:
5418:
5289:
5257:
5092:
5064:
5038:
5009:
4983:
4960:
4802:
4696:
4676:
4656:
4621:
4586:
4551:
4513:
4413:
4378:
4257:
4228:
4186:
4045:
4022:
4002:
3976:
3869:
3628:
3601:
3496:
3255:
3226:
3053:
3033:
3006:
2979:
2926:
2900:
2868:
2848:
2816:
2731:
2681:
2661:
2641:
2600:
2566:
2546:
2524:
2496:
2469:
2442:
2409:
2389:
2363:
2333:
2302:
2282:
2212:
2192:
2172:
2145:
2114:
2094:
2061:
2041:
2005:
1988:
1977:
1948:
1919:
1887:
1784:
1758:
1738:
1702:
1663:
1642:
1614:
1585:
1556:
1524:
1424:
1398:
1378:
1342:
1292:
1198:
1037:
1001:
962:
889:
846:
815:
718:
682:
615:
584:
564:
526:
497:
468:
439:
410:
321:
301:
274:
244:
165:
138:
109:
79:
6315:
6231:
5429:
592:is a circle, the problem reduces to deriving the
7360:
6980:
6123:
4423:). It is the complex form of the nonhomogeneous
1220:
1045:on the inside and outside, respectively, of the
963:{\displaystyle M_{-}(z)={\overline {M_{+}(z)}}.}
6835:
6150:
6134:
4053:is crucial. Otherwise any function of the form
6940:Communications on Pure and Applied Mathematics
6886:
6722:
6662:
6516:Deift, Percy; Venakides, S.; Zhou, X. (1997),
6479:
6401:Communications on Pure and Applied Mathematics
6201:
6165:
5987:The most celebrated example is the theorem of
5969:
5933:or inverse spectral problem associated to the
5903:, in solving certain type of multidimensional
7116:
6349:
6333:
6309:
6297:
6285:
6249:
6213:
119:. Divide the plane into two parts denoted by
7158:Grothendieck–Hirzebruch–Riemann–Roch theorem
7069:
6933:
6397:
6376:Journal of the American Mathematical Society
6092:
6065:
6064:and using technical background results from
6028:
4664:all real-valued functions of real variables
677:
641:
6369:Baik, J.; Deift, P.; Johansson, K. (1999),
6133:Another extension of the theory appears in
5953:can be stated as a Riemann–Hilbert problem.
1625:
7123:
7109:
6996:. New York, N.Y: Taylor & Francis US.
6495:Orthogonal Polynomials and Random Matrices
6438:
6012:
7303:Riemann–Roch theorem for smooth manifolds
6900:
6853:
6802:
6544:
6525:
6454:
6388:
6354:. Cambridge: Cambridge University Press.
6113:Kamvissis, McLaughlin & Miller (2003)
6053:
5277:
5227:
5171:
5086:
3205:
3132:
840:
651:
634:
7070:Trogdon, Thomas; Olver, Sheehan (2016),
7020:
6420:
6350:Ablowitz, Mark J.; Fokas, A. S. (2003).
6225:
6127:
5905:nonlinear partial differential equations
603:, it suffices to consider the case when
6959:
6061:
4414:{\displaystyle {\overline {\partial }}}
2290:, at all points of non-intersection in
411:{\displaystyle a(t)u(t)-b(t)v(t)=c(t),}
7361:
7010:
6701:
6666:Communications in Mathematical Physics
6627:Communications in Mathematical Physics
6588:Communications in Mathematical Physics
6261:
6237:
6080:; also cf. the scalar example below).
6077:
6034:
5463:is holomorphic in some complex region
5100:, the solution of the DBAR problem is
3265:Liouville's theorem (complex analysis)
1303:
62:
7104:
7050:
6991:
6623:
6577:
6489:
6321:
6273:
6048:applicable to exponential integrals.
5997:
5973:
5942:
2283:{\displaystyle M_{+}(t)=G(t)M_{-}(t)}
2015:Riemann–Hilbert factorization problem
7130:
7079:
6069:
5915:Applications to integrability theory
2180:denote the non-tangential limits of
1985:are given complex-valued functions.
245:{\displaystyle M_{+}(t)=u(t)+iv(t),}
6755:
6741:
6057:
282:, such that the boundary values of
13:
7268:Riemannian connection on a surface
7173:Measurable Riemann mapping theorem
6085:Deift, Venakides & Zhou (1997)
6056:, expanding on a previous idea by
5989:Baik, Deift & Johansson (1999)
5585:
5577:
5504:
5496:
5059:
4925:
4917:
4902:
4894:
4857:
4849:
4834:
4826:
4783:
4779:
4762:
4758:
4720:
4716:
4403:
4312:
4280:
4200:
3623:
3316:
3250:
3166:
3095:
3048:
2921:
2615:
2541:
2404:
2384:
2297:
2207:
2109:
2056:
2000:
1779:
1753:
1658:
1637:
1419:
1393:
1230:
610:
579:
434:
316:
263:
154:
127:
74:
14:
7435:
6480:Clancey, K.; Gohberg, I. (1981),
2669:is bounded, what is the solution
847:{\displaystyle z\in \mathbb {T} }
173:(the outside), determined by the
7341:
7340:
6947:(3): 253–290, 571–593, 809–829,
6117:Kamvissis & Rakhmanov (2005)
5290:{\displaystyle \mathbb {R} ^{2}}
7404:Ordinary differential equations
7253:Riemann's differential equation
7163:Hirzebruch–Riemann–Roch theorem
7024:Journal of Mathematical Physics
6790:Journal of Mathematical Physics
5840:{\displaystyle B(z,{\bar {z}})}
5796:{\displaystyle A(z,{\bar {z}})}
5732:{\displaystyle {\overline {M}}}
5093:{\displaystyle D:=\mathbb {C} }
4890:
4489:
4458:
4363:
4205:
1989:Matrix Riemann–Hilbert problems
7409:Partial differential equations
7278:Riemann–Hilbert correspondence
7148:Generalized Riemann hypothesis
7053:Generalized Analytic Functions
6178:Riemann–Hilbert correspondence
6124:McLaughlin & Miller (2006)
5970:Fokas, Its & Kitaev (1992)
5939:partial differential equations
5880:
5834:
5828:
5813:
5790:
5784:
5769:
5663:
5657:
5642:
5630:
5624:
5609:
5594:
5513:
5450:
5444:
5430:Generalized analytic functions
5386:
5365:
5359:
5338:
5329:
5246:
5210:
5204:
5189:
5134:
5128:
5113:
5056:
5030:
4952:
4940:
4884:
4872:
4729:
4651:
4639:
4616:
4604:
4581:
4569:
4546:
4534:
4357:
4351:
4336:
4321:
4307:
4301:
4286:
4072:
4066:
3968:
3962:
3928:
3922:
3906:
3900:
3802:
3782:
3772:
3766:
3696:
3679:
3669:
3663:
3526:
3520:
2918:
2892:
2843:
2837:
2799:
2793:
2771:
2765:
2692:To solve this, let's take the
2636:
2621:
2437:
2431:
2381:
2328:
2322:
2277:
2271:
2258:
2252:
2243:
2237:
2089:
2083:
2036:
2030:
1972:
1966:
1943:
1937:
1914:
1908:
1879:
1873:
1864:
1858:
1845:
1839:
1830:
1824:
1811:
1805:
1766:, respectively, such that for
1733:
1727:
1697:
1691:
1609:
1603:
1580:
1574:
1551:
1545:
1519:
1513:
1504:
1498:
1485:
1479:
1470:
1464:
1451:
1445:
1373:
1367:
1337:
1331:
1278:
1272:
1251:
1245:
1227:
1190:
1184:
1175:
1169:
1150:
1144:
1132:
1126:
1114:
1108:
1089:
1083:
1071:
1065:
1032:
1026:
996:
990:
948:
942:
923:
917:
890:{\displaystyle z=1/{\bar {z}}}
881:
785:
755:
749:
713:
707:
667:
659:
521:
515:
492:
486:
463:
457:
402:
396:
387:
381:
375:
369:
360:
354:
348:
342:
236:
230:
218:
212:
203:
197:
18:Hilbert's twenty-first problem
1:
7313:Riemann–Siegel theta function
6499:American Mathematical Society
6390:10.1090/S0894-0347-99-00307-0
6343:
6151:Kamvissis & Teschl (2012)
6135:Kamvissis & Teschl (2012)
2980:{\displaystyle C_{+}-C_{-}=I}
2102:defined on the complement of
1049:, so that on the unit circle
690:. In this case, one may seek
7328:Riemann–von Mangoldt formula
6744:Soviet Mathematics - Doklady
6723:Khimshiashvili, G. (2001) ,
6166:Kuijlaars & López (2015)
6164:and collaborators, see e.g.
5739:is the complex conjugate of
5724:
5671:
5065:{\displaystyle z\to \infty }
4406:
3256:{\displaystyle z\in \Sigma }
2927:{\displaystyle z\to \infty }
2732:{\displaystyle M_{+}=GM_{-}}
2397:along any direction outside
2390:{\displaystyle z\to \infty }
1785:{\displaystyle t\in \Sigma }
1425:{\displaystyle t\in \Sigma }
1282:
952:
805:
440:{\displaystyle t\in \Sigma }
7:
6730:Encyclopedia of Mathematics
6712:Encyclopedia of Mathematics
6428:Encyclopedia of Mathematics
6171:
4267:. Then the scalar equation
3271:gives the solution. We get
2901:{\displaystyle \log M\to 0}
1746:on the "+" and "−" side of
275:{\displaystyle \Sigma _{+}}
166:{\displaystyle \Sigma _{-}}
139:{\displaystyle \Sigma _{+}}
10:
7440:
7323:Riemann–Stieltjes integral
7318:Riemann–Silberstein vector
7293:Riemann–Liouville integral
6919:10.1088/0951-7715/28/2/347
6202:Clancey & Gohberg 1981
6093:Lax & Levermore (1983)
6066:Beals & Coifman (1984)
6046:method of steepest descent
6042:method of stationary phase
6029:Trogdon & Olver (2016)
5889:{\displaystyle {\bar {z}}}
4209:
2937:A standard fact about the
1993:Given an oriented contour
1978:{\displaystyle \gamma (t)}
1920:{\displaystyle \alpha (t)}
1615:{\displaystyle \gamma (t)}
1557:{\displaystyle \alpha (t)}
1307:
15:
7336:
7258:Riemann's minimal surface
7138:
6707:"Riemann–Hilbert problem"
6648:10.1007/s00220-002-0681-8
6473:10.1007/s00222-018-0843-8
6334:Ablowitz & Fokas 2003
6310:Ablowitz & Fokas 2003
6298:Ablowitz & Fokas 2003
6286:Ablowitz & Fokas 2003
6250:Ablowitz & Fokas 2003
6214:Ablowitz & Fokas 2003
3269:Sokhotski–Plemelj theorem
1949:{\displaystyle \beta (t)}
1586:{\displaystyle \beta (t)}
7283:Riemann–Hilbert problems
7188:Riemann curvature tensor
7153:Grand Riemann hypothesis
7143:Cauchy–Riemann equations
6725:"Birkhoff factorization"
6442:Inventiones Mathematicae
6421:Bitsadze, A.V. (2001) ,
6188:
4813:the DBAR problem yields
4425:Cauchy-Riemann equations
4010:near the special points
2642:{\displaystyle \Sigma =}
2020:Given a matrix function
1739:{\displaystyle M_{-}(t)}
1703:{\displaystyle M_{+}(t)}
1626:Riemann–Hilbert problems
1379:{\displaystyle M_{-}(t)}
1343:{\displaystyle M_{+}(t)}
1038:{\displaystyle M_{-}(z)}
1002:{\displaystyle M_{+}(z)}
719:{\displaystyle M_{+}(z)}
29:Riemann–Hilbert problems
7374:Exactly solvable models
7208:Riemann mapping theorem
6054:Deift & Zhou (1993)
6011:The work of Bridgeland
6007:Donaldson-Thomas theory
3629:{\displaystyle \Sigma }
3054:{\displaystyle \Sigma }
2547:{\displaystyle \Sigma }
2410:{\displaystyle \Sigma }
2303:{\displaystyle \Sigma }
2213:{\displaystyle \Sigma }
2115:{\displaystyle \Sigma }
2062:{\displaystyle \Sigma }
2049:defined on the contour
2006:{\displaystyle \Sigma }
1759:{\displaystyle \Sigma }
1664:{\displaystyle \Sigma }
1643:{\displaystyle \Sigma }
1399:{\displaystyle \Sigma }
616:{\displaystyle \Sigma }
601:Riemann mapping theorem
585:{\displaystyle \Sigma }
565:{\displaystyle a=1,b=0}
322:{\displaystyle \Sigma }
80:{\displaystyle \Sigma }
7308:Riemann–Siegel formula
7288:Riemann–Lebesgue lemma
7223:Riemann series theorem
6953:10.1002/cpa.3160360302
6414:10.1002/cpa.3160370105
5960:Orthogonal polynomials
5890:
5861:
5841:
5797:
5753:
5733:
5706:
5683:
5558:
5535:
5477:
5457:
5420:
5291:
5259:
5094:
5066:
5040:
5039:{\displaystyle M\to 1}
5011:
5010:{\displaystyle z\in D}
4985:
4962:
4804:
4698:
4678:
4658:
4657:{\displaystyle h(x,y)}
4623:
4622:{\displaystyle g(x,y)}
4588:
4587:{\displaystyle v(x,y)}
4553:
4552:{\displaystyle u(x,y)}
4515:
4415:
4380:
4259:
4230:
4188:
4047:
4024:
4004:
3978:
3871:
3630:
3603:
3498:
3257:
3228:
3055:
3035:
3008:
2981:
2928:
2902:
2870:
2850:
2818:
2733:
2683:
2663:
2643:
2602:
2568:
2548:
2526:
2498:
2471:
2444:
2411:
2391:
2365:
2335:
2304:
2284:
2214:
2194:
2174:
2147:
2116:
2096:
2063:
2043:
2007:
1979:
1950:
1921:
1889:
1786:
1760:
1740:
1704:
1665:
1644:
1616:
1587:
1558:
1526:
1426:
1400:
1380:
1344:
1294:
1200:
1039:
1003:
964:
891:
848:
817:
720:
684:
617:
586:
566:
528:
499:
470:
441:
412:
323:
303:
276:
246:
167:
140:
111:
93:simple, closed contour
81:
41:differential equations
7414:Mathematical problems
7248:Riemann zeta function
7051:Vekua, I. N. (2014).
7011:Pandey, J.N. (1996),
6532:Annals of Mathematics
5907:and multidimensional
5901:differential geometry
5891:
5862:
5842:
5798:
5754:
5734:
5707:
5684:
5559:
5536:
5478:
5458:
5421:
5292:
5260:
5095:
5067:
5041:
5012:
4986:
4963:
4805:
4699:
4679:
4659:
4624:
4589:
4554:
4516:
4416:
4381:
4260:
4231:
4189:
4048:
4025:
4005:
3979:
3872:
3631:
3604:
3499:
3258:
3229:
3056:
3036:
3034:{\displaystyle C_{-}}
3009:
3007:{\displaystyle C_{+}}
2982:
2929:
2903:
2871:
2851:
2819:
2734:
2684:
2664:
2644:
2603:
2569:
2549:
2527:
2525:{\displaystyle L^{2}}
2499:
2497:{\displaystyle M_{-}}
2472:
2470:{\displaystyle M_{+}}
2445:
2421:In the simplest case
2412:
2392:
2366:
2364:{\displaystyle I_{N}}
2336:
2305:
2285:
2215:
2195:
2175:
2173:{\displaystyle M_{-}}
2148:
2146:{\displaystyle M_{+}}
2117:
2097:
2064:
2044:
2008:
1980:
1951:
1922:
1890:
1787:
1761:
1741:
1705:
1666:
1645:
1617:
1588:
1559:
1527:
1427:
1401:
1381:
1345:
1295:
1201:
1040:
1004:
965:
892:
849:
818:
721:
685:
618:
587:
567:
529:
500:
471:
442:
413:
329:satisfy the equation
324:
304:
302:{\displaystyle M_{+}}
277:
247:
168:
141:
112:
82:
7298:Riemann–Roch theorem
7015:, Wiley-Interscience
6147:Riemann–Roch theorem
5937:for 1+1 dimensional
5871:
5851:
5807:
5763:
5743:
5716:
5696:
5571:
5548:
5490:
5467:
5456:{\displaystyle M(z)}
5438:
5308:
5272:
5107:
5076:
5050:
5024:
4995:
4975:
4820:
4711:
4688:
4668:
4633:
4598:
4563:
4528:
4434:
4427:. To show this, let
4398:
4274:
4249:
4220:
4060:
4034:
4014:
3994:
3887:
3646:
3620:
3514:
3278:
3241:
3068:
3061:; therefore, we get
3045:
3018:
2991:
2945:
2912:
2880:
2860:
2849:{\displaystyle M(z)}
2831:
2746:
2700:
2673:
2653:
2612:
2586:
2558:
2538:
2509:
2481:
2454:
2443:{\displaystyle G(t)}
2425:
2401:
2375:
2348:
2334:{\displaystyle M(z)}
2316:
2294:
2224:
2204:
2184:
2157:
2130:
2106:
2095:{\displaystyle M(z)}
2077:
2053:
2042:{\displaystyle G(t)}
2024:
1997:
1960:
1931:
1902:
1799:
1770:
1750:
1714:
1678:
1655:
1634:
1597:
1568:
1539:
1439:
1410:
1390:
1354:
1318:
1216:
1056:
1013:
977:
904:
858:
830:
736:
694:
630:
607:
576:
538:
527:{\displaystyle c(t)}
509:
498:{\displaystyle b(t)}
480:
469:{\displaystyle a(t)}
451:
425:
336:
313:
286:
259:
184:
150:
123:
101:
71:
7399:Microlocal analysis
7273:Riemannian geometry
7183:Riemann Xi function
7168:Local zeta function
7082:SIAM J. Math. Anal.
7037:1996JMP....37.5869V
6992:Noble, Ben (1958).
6974:1974JETP...38..693M
6911:2015Nonli..28..347K
6864:2012JMP....53g3706K
6813:2005JMP....46h3505K
6679:1992CMaPh.147..395F
6640:2002CMaPh.230....1F
6601:1976CMaPh..47..171D
6465:2019InMat.216...69B
6145:, by reason of the
6035:Use for asymptotics
4991:is holomorphic for
3413:
2601:{\displaystyle G=2}
1304:The Hilbert problem
63:The Riemann problem
7379:Integrable systems
7193:Riemann hypothesis
6765:Notices of the AMS
6756:Its, A.R. (2003),
6687:10.1007/BF02096594
6609:10.1007/BF01608375
6183:Wiener-Hopf method
5951:Painlevé equations
5931:inverse scattering
5909:inverse scattering
5886:
5857:
5837:
5793:
5749:
5729:
5702:
5679:
5554:
5531:
5473:
5453:
5416:
5287:
5255:
5090:
5062:
5036:
5007:
4981:
4958:
4800:
4694:
4674:
4654:
4619:
4584:
4549:
4511:
4411:
4376:
4255:
4226:
4184:
4046:{\displaystyle -1}
4043:
4020:
4000:
3974:
3867:
3865:
3626:
3599:
3494:
3384:
3253:
3224:
3051:
3031:
3004:
2977:
2924:
2898:
2866:
2846:
2814:
2729:
2679:
2659:
2639:
2598:
2564:
2544:
2522:
2494:
2467:
2440:
2407:
2387:
2361:
2331:
2300:
2280:
2210:
2190:
2170:
2143:
2112:
2092:
2059:
2039:
2017:is the following.
2003:
1975:
1946:
1917:
1885:
1782:
1756:
1736:
1700:
1673:analytic functions
1661:
1640:
1612:
1583:
1554:
1522:
1422:
1396:
1376:
1340:
1310:Wiener-Hopf method
1290:
1234:
1196:
1035:
999:
960:
887:
844:
813:
728:Schwarz reflection
716:
680:
613:
582:
562:
524:
495:
466:
437:
408:
319:
299:
272:
242:
163:
136:
107:
77:
49:existence theorems
7394:Harmonic analysis
7389:Scattering theory
7356:
7355:
7263:Riemannian circle
7203:Riemann invariant
7062:978-1-61427-611-1
7031:(11): 5869–5892,
7003:978-0-8284-0332-0
6872:10.1063/1.4731768
6821:10.1063/1.1985069
6535:, Second Series,
6508:978-0-8218-2695-9
6361:978-0-521-53429-1
6352:Complex Variables
6216:, pp. 71–72.
6013:Bridgeland (2019)
6005:D. Connection to
5981:C. Combinatorial
5925:Integrable models
5883:
5860:{\displaystyle z}
5847:are functions of
5831:
5787:
5752:{\displaystyle M}
5727:
5705:{\displaystyle R}
5674:
5660:
5627:
5601:
5597:
5557:{\displaystyle R}
5520:
5516:
5476:{\displaystyle R}
5332:
5249:
5225:
5207:
5162:
5131:
4984:{\displaystyle M}
4932:
4909:
4864:
4841:
4790:
4769:
4749:
4736:
4732:
4697:{\displaystyle y}
4677:{\displaystyle x}
4484:
4409:
4354:
4328:
4324:
4304:
4258:{\displaystyle z}
4229:{\displaystyle D}
4182:
4161:
4139:
4107:
4023:{\displaystyle 1}
4003:{\displaystyle M}
3859:
3829:
3749:
3723:
3593:
3561:
3489:
3457:
3422:
3382:
3347:
3309:
3203:
3158:
3130:
3087:
2869:{\displaystyle 1}
2682:{\displaystyle M}
2662:{\displaystyle M}
2567:{\displaystyle M}
2193:{\displaystyle M}
1285:
1219:
1157:
1096:
955:
884:
808:
788:
146:(the inside) and
110:{\displaystyle z}
7431:
7419:Bernhard Riemann
7369:Complex analysis
7344:
7343:
7198:Riemann integral
7178:Riemann (crater)
7132:Bernhard Riemann
7125:
7118:
7111:
7102:
7101:
7096:
7075:
7066:
7047:
7045:10.1063/1.531706
7016:
7007:
6987:
6976:
6955:
6929:
6904:
6882:
6857:
6831:
6806:
6782:
6772:
6762:
6751:
6737:
6719:
6697:
6658:
6619:
6573:
6548:
6521:
6511:
6485:
6475:
6458:
6435:
6416:
6393:
6392:
6383:(4): 1119–1178,
6365:
6337:
6331:
6325:
6319:
6313:
6307:
6301:
6295:
6289:
6283:
6277:
6271:
6265:
6259:
6253:
6247:
6241:
6235:
6229:
6223:
6217:
6211:
6205:
6199:
5895:
5893:
5892:
5887:
5885:
5884:
5876:
5866:
5864:
5863:
5858:
5846:
5844:
5843:
5838:
5833:
5832:
5824:
5802:
5800:
5799:
5794:
5789:
5788:
5780:
5758:
5756:
5755:
5750:
5738:
5736:
5735:
5730:
5728:
5720:
5711:
5709:
5708:
5703:
5688:
5686:
5685:
5680:
5675:
5667:
5662:
5661:
5653:
5629:
5628:
5620:
5602:
5600:
5599:
5598:
5590:
5583:
5575:
5563:
5561:
5560:
5555:
5540:
5538:
5537:
5532:
5521:
5519:
5518:
5517:
5509:
5502:
5494:
5482:
5480:
5479:
5474:
5462:
5460:
5459:
5454:
5425:
5423:
5422:
5417:
5334:
5333:
5325:
5297:, and where the
5296:
5294:
5293:
5288:
5286:
5285:
5280:
5264:
5262:
5261:
5256:
5251:
5250:
5242:
5226:
5224:
5213:
5209:
5208:
5200:
5184:
5182:
5181:
5180:
5179:
5174:
5163:
5161:
5147:
5133:
5132:
5124:
5099:
5097:
5096:
5091:
5089:
5071:
5069:
5068:
5063:
5045:
5043:
5042:
5037:
5016:
5014:
5013:
5008:
4990:
4988:
4987:
4982:
4967:
4965:
4964:
4959:
4933:
4931:
4923:
4915:
4910:
4908:
4900:
4892:
4865:
4863:
4855:
4847:
4842:
4840:
4832:
4824:
4809:
4807:
4806:
4801:
4796:
4792:
4791:
4789:
4778:
4770:
4768:
4757:
4750:
4742:
4737:
4735:
4734:
4733:
4725:
4715:
4703:
4701:
4700:
4695:
4683:
4681:
4680:
4675:
4663:
4661:
4660:
4655:
4628:
4626:
4625:
4620:
4593:
4591:
4590:
4585:
4558:
4556:
4555:
4550:
4520:
4518:
4517:
4512:
4485:
4480:
4466:
4420:
4418:
4417:
4412:
4410:
4402:
4385:
4383:
4382:
4377:
4356:
4355:
4347:
4329:
4327:
4326:
4325:
4317:
4310:
4306:
4305:
4297:
4278:
4264:
4262:
4261:
4256:
4238:simply connected
4235:
4233:
4232:
4227:
4193:
4191:
4190:
4185:
4183:
4181:
4167:
4162:
4160:
4146:
4141:
4140:
4138:
4127:
4126:
4114:
4112:
4108:
4106:
4095:
4084:
4052:
4050:
4049:
4044:
4029:
4027:
4026:
4021:
4009:
4007:
4006:
4001:
3983:
3981:
3980:
3975:
3961:
3960:
3948:
3947:
3946:
3921:
3920:
3899:
3898:
3876:
3874:
3873:
3868:
3866:
3862:
3861:
3860:
3855:
3844:
3831:
3830:
3828:
3817:
3806:
3800:
3799:
3765:
3764:
3751:
3750:
3745:
3734:
3725:
3724:
3722:
3711:
3700:
3694:
3693:
3662:
3661:
3635:
3633:
3632:
3627:
3608:
3606:
3605:
3600:
3595:
3594:
3592:
3581:
3580:
3568:
3566:
3562:
3560:
3549:
3538:
3503:
3501:
3500:
3495:
3490:
3488:
3477:
3466:
3458:
3456:
3445:
3434:
3423:
3415:
3412:
3401:
3383:
3381:
3370:
3359:
3348:
3346:
3335:
3334:
3322:
3320:
3319:
3310:
3308:
3294:
3262:
3260:
3259:
3254:
3233:
3231:
3230:
3225:
3204:
3202:
3191:
3190:
3178:
3176:
3175:
3174:
3173:
3159:
3157:
3143:
3131:
3129:
3118:
3107:
3105:
3104:
3103:
3102:
3088:
3086:
3072:
3060:
3058:
3057:
3052:
3040:
3038:
3037:
3032:
3030:
3029:
3013:
3011:
3010:
3005:
3003:
3002:
2986:
2984:
2983:
2978:
2970:
2969:
2957:
2956:
2939:Cauchy transform
2933:
2931:
2930:
2925:
2907:
2905:
2904:
2899:
2875:
2873:
2872:
2867:
2855:
2853:
2852:
2847:
2823:
2821:
2820:
2815:
2792:
2791:
2764:
2763:
2738:
2736:
2735:
2730:
2728:
2727:
2712:
2711:
2688:
2686:
2685:
2680:
2668:
2666:
2665:
2660:
2648:
2646:
2645:
2640:
2607:
2605:
2604:
2599:
2573:
2571:
2570:
2565:
2553:
2551:
2550:
2545:
2531:
2529:
2528:
2523:
2521:
2520:
2503:
2501:
2500:
2495:
2493:
2492:
2476:
2474:
2473:
2468:
2466:
2465:
2449:
2447:
2446:
2441:
2416:
2414:
2413:
2408:
2396:
2394:
2393:
2388:
2370:
2368:
2367:
2362:
2360:
2359:
2340:
2338:
2337:
2332:
2309:
2307:
2306:
2301:
2289:
2287:
2286:
2281:
2270:
2269:
2236:
2235:
2219:
2217:
2216:
2211:
2199:
2197:
2196:
2191:
2179:
2177:
2176:
2171:
2169:
2168:
2152:
2150:
2149:
2144:
2142:
2141:
2121:
2119:
2118:
2113:
2101:
2099:
2098:
2093:
2073:matrix function
2068:
2066:
2065:
2060:
2048:
2046:
2045:
2040:
2012:
2010:
2009:
2004:
1984:
1982:
1981:
1976:
1955:
1953:
1952:
1947:
1926:
1924:
1923:
1918:
1894:
1892:
1891:
1886:
1857:
1856:
1823:
1822:
1791:
1789:
1788:
1783:
1765:
1763:
1762:
1757:
1745:
1743:
1742:
1737:
1726:
1725:
1709:
1707:
1706:
1701:
1690:
1689:
1670:
1668:
1667:
1662:
1649:
1647:
1646:
1641:
1621:
1619:
1618:
1613:
1592:
1590:
1589:
1584:
1563:
1561:
1560:
1555:
1531:
1529:
1528:
1523:
1497:
1496:
1463:
1462:
1431:
1429:
1428:
1423:
1406:, such that for
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1366:
1365:
1349:
1347:
1346:
1341:
1330:
1329:
1299:
1297:
1296:
1291:
1286:
1281:
1271:
1270:
1265:
1258:
1244:
1243:
1233:
1205:
1203:
1202:
1197:
1168:
1167:
1158:
1153:
1121:
1107:
1106:
1097:
1092:
1060:
1044:
1042:
1041:
1036:
1025:
1024:
1008:
1006:
1005:
1000:
989:
988:
969:
967:
966:
961:
956:
951:
941:
940:
930:
916:
915:
896:
894:
893:
888:
886:
885:
877:
874:
853:
851:
850:
845:
843:
822:
820:
819:
814:
809:
804:
803:
799:
798:
790:
789:
781:
773:
772:
762:
748:
747:
725:
723:
722:
717:
706:
705:
689:
687:
686:
681:
670:
662:
654:
637:
622:
620:
619:
614:
591:
589:
588:
583:
571:
569:
568:
563:
533:
531:
530:
525:
504:
502:
501:
496:
475:
473:
472:
467:
446:
444:
443:
438:
417:
415:
414:
409:
328:
326:
325:
320:
308:
306:
305:
300:
298:
297:
281:
279:
278:
273:
271:
270:
255:analytic inside
251:
249:
248:
243:
196:
195:
172:
170:
169:
164:
162:
161:
145:
143:
142:
137:
135:
134:
116:
114:
113:
108:
86:
84:
83:
78:
33:Bernhard Riemann
7439:
7438:
7434:
7433:
7432:
7430:
7429:
7428:
7359:
7358:
7357:
7352:
7332:
7243:Riemann surface
7218:Riemann problem
7134:
7129:
7094:10.1137/0520065
7063:
7004:
6962:Sov. Phys. JETP
6836:Kamvissis, S.;
6771:(11): 1389–1400
6760:
6555:10.2307/2946540
6509:
6491:Deift, Percy A.
6362:
6346:
6341:
6340:
6332:
6328:
6320:
6316:
6308:
6304:
6296:
6292:
6284:
6280:
6272:
6268:
6260:
6256:
6252:, pp. 514.
6248:
6244:
6236:
6232:
6224:
6220:
6212:
6208:
6200:
6196:
6191:
6174:
6139:Riemann surface
6128:Varzugin (1996)
6037:
5964:Random matrices
5935:Cauchy problems
5917:
5875:
5874:
5872:
5869:
5868:
5852:
5849:
5848:
5823:
5822:
5808:
5805:
5804:
5779:
5778:
5764:
5761:
5760:
5744:
5741:
5740:
5719:
5717:
5714:
5713:
5697:
5694:
5693:
5666:
5652:
5651:
5619:
5618:
5589:
5588:
5584:
5576:
5574:
5572:
5569:
5568:
5549:
5546:
5545:
5508:
5507:
5503:
5495:
5493:
5491:
5488:
5487:
5468:
5465:
5464:
5439:
5436:
5435:
5432:
5324:
5323:
5309:
5306:
5305:
5281:
5276:
5275:
5273:
5270:
5269:
5241:
5240:
5214:
5199:
5198:
5185:
5183:
5175:
5170:
5169:
5168:
5164:
5151:
5146:
5123:
5122:
5108:
5105:
5104:
5085:
5077:
5074:
5073:
5051:
5048:
5047:
5025:
5022:
5021:
4996:
4993:
4992:
4976:
4973:
4972:
4924:
4916:
4914:
4901:
4893:
4891:
4856:
4848:
4846:
4833:
4825:
4823:
4821:
4818:
4817:
4782:
4777:
4761:
4756:
4755:
4751:
4741:
4724:
4723:
4719:
4714:
4712:
4709:
4708:
4689:
4686:
4685:
4669:
4666:
4665:
4634:
4631:
4630:
4599:
4596:
4595:
4564:
4561:
4560:
4529:
4526:
4525:
4467:
4465:
4435:
4432:
4431:
4401:
4399:
4396:
4395:
4346:
4345:
4316:
4315:
4311:
4296:
4295:
4279:
4277:
4275:
4272:
4271:
4250:
4247:
4246:
4221:
4218:
4217:
4214:
4208:
4203:
4201:Generalizations
4171:
4166:
4150:
4145:
4128:
4122:
4115:
4113:
4096:
4085:
4083:
4079:
4078:
4061:
4058:
4057:
4035:
4032:
4031:
4015:
4012:
4011:
3995:
3992:
3991:
3956:
3952:
3942:
3935:
3931:
3916:
3912:
3894:
3890:
3888:
3885:
3884:
3864:
3863:
3845:
3843:
3839:
3835:
3818:
3807:
3805:
3801:
3789:
3785:
3775:
3760:
3756:
3753:
3752:
3735:
3733:
3729:
3712:
3701:
3699:
3695:
3686:
3682:
3672:
3657:
3653:
3649:
3647:
3644:
3643:
3621:
3618:
3617:
3582:
3576:
3569:
3567:
3550:
3539:
3537:
3533:
3532:
3515:
3512:
3511:
3478:
3467:
3465:
3446:
3435:
3433:
3414:
3402:
3388:
3371:
3360:
3358:
3336:
3330:
3323:
3321:
3315:
3311:
3298:
3293:
3279:
3276:
3275:
3242:
3239:
3238:
3192:
3186:
3179:
3177:
3169:
3165:
3164:
3160:
3147:
3142:
3119:
3108:
3106:
3098:
3094:
3093:
3089:
3076:
3071:
3069:
3066:
3065:
3046:
3043:
3042:
3025:
3021:
3019:
3016:
3015:
2998:
2994:
2992:
2989:
2988:
2965:
2961:
2952:
2948:
2946:
2943:
2942:
2913:
2910:
2909:
2881:
2878:
2877:
2861:
2858:
2857:
2832:
2829:
2828:
2787:
2783:
2759:
2755:
2747:
2744:
2743:
2723:
2719:
2707:
2703:
2701:
2698:
2697:
2674:
2671:
2670:
2654:
2651:
2650:
2613:
2610:
2609:
2587:
2584:
2583:
2580:
2559:
2556:
2555:
2539:
2536:
2535:
2516:
2512:
2510:
2507:
2506:
2488:
2484:
2482:
2479:
2478:
2461:
2457:
2455:
2452:
2451:
2426:
2423:
2422:
2402:
2399:
2398:
2376:
2373:
2372:
2355:
2351:
2349:
2346:
2345:
2343:identity matrix
2317:
2314:
2313:
2295:
2292:
2291:
2265:
2261:
2231:
2227:
2225:
2222:
2221:
2205:
2202:
2201:
2200:as we approach
2185:
2182:
2181:
2164:
2160:
2158:
2155:
2154:
2137:
2133:
2131:
2128:
2127:
2107:
2104:
2103:
2078:
2075:
2074:
2054:
2051:
2050:
2025:
2022:
2021:
1998:
1995:
1994:
1991:
1961:
1958:
1957:
1932:
1929:
1928:
1903:
1900:
1899:
1852:
1848:
1818:
1814:
1800:
1797:
1796:
1771:
1768:
1767:
1751:
1748:
1747:
1721:
1717:
1715:
1712:
1711:
1685:
1681:
1679:
1676:
1675:
1656:
1653:
1652:
1635:
1632:
1631:
1628:
1598:
1595:
1594:
1569:
1566:
1565:
1540:
1537:
1536:
1492:
1488:
1458:
1454:
1440:
1437:
1436:
1411:
1408:
1407:
1391:
1388:
1387:
1361:
1357:
1355:
1352:
1351:
1325:
1321:
1319:
1316:
1315:
1312:
1306:
1266:
1261:
1260:
1259:
1257:
1239:
1235:
1223:
1217:
1214:
1213:
1163:
1159:
1122:
1120:
1102:
1098:
1061:
1059:
1057:
1054:
1053:
1020:
1016:
1014:
1011:
1010:
984:
980:
978:
975:
974:
936:
932:
931:
929:
911:
907:
905:
902:
901:
876:
875:
870:
859:
856:
855:
839:
831:
828:
827:
791:
780:
779:
778:
774:
768:
764:
763:
761:
743:
739:
737:
734:
733:
726:along with its
701:
697:
695:
692:
691:
666:
658:
650:
633:
631:
628:
627:
608:
605:
604:
594:Poisson formula
577:
574:
573:
539:
536:
535:
510:
507:
506:
481:
478:
477:
452:
449:
448:
426:
423:
422:
337:
334:
333:
314:
311:
310:
293:
289:
287:
284:
283:
266:
262:
260:
257:
256:
191:
187:
185:
182:
181:
157:
153:
151:
148:
147:
130:
126:
124:
121:
120:
102:
99:
98:
72:
69:
68:
65:
21:
12:
11:
5:
7437:
7427:
7426:
7421:
7416:
7411:
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7354:
7353:
7351:
7350:
7337:
7334:
7333:
7331:
7330:
7325:
7320:
7315:
7310:
7305:
7300:
7295:
7290:
7285:
7280:
7275:
7270:
7265:
7260:
7255:
7250:
7245:
7240:
7235:
7233:Riemann sphere
7230:
7228:Riemann solver
7225:
7220:
7215:
7210:
7205:
7200:
7195:
7190:
7185:
7180:
7175:
7170:
7165:
7160:
7155:
7150:
7145:
7139:
7136:
7135:
7128:
7127:
7120:
7113:
7105:
7099:
7098:
7088:(4): 966–986,
7077:
7067:
7061:
7048:
7018:
7008:
7002:
6989:
6978:
6957:
6931:
6895:(2): 347–406,
6884:
6842:J. Math. Phys.
6833:
6784:
6774:
6753:
6739:
6720:
6699:
6673:(2): 395–430,
6660:
6621:
6595:(3): 171–183,
6579:Dyson, Freeman
6575:
6539:(2): 295–368,
6523:
6513:
6507:
6487:
6477:
6436:
6418:
6395:
6366:
6360:
6345:
6342:
6339:
6338:
6336:, p. 601.
6326:
6314:
6312:, p. 600.
6302:
6300:, p. 598.
6290:
6278:
6266:
6254:
6242:
6230:
6218:
6206:
6193:
6192:
6190:
6187:
6186:
6185:
6180:
6173:
6170:
6162:Arno Kuijlaars
6062:Manakov (1974)
6036:
6033:
6021:
6020:
6009:
6002:
6001:
5985:
5978:
5977:
5966:
5955:
5954:
5927:
5916:
5913:
5882:
5879:
5856:
5836:
5830:
5827:
5821:
5818:
5815:
5812:
5792:
5786:
5783:
5777:
5774:
5771:
5768:
5748:
5726:
5723:
5701:
5690:
5689:
5678:
5673:
5670:
5665:
5659:
5656:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5626:
5623:
5617:
5614:
5611:
5608:
5605:
5596:
5593:
5587:
5582:
5579:
5553:
5542:
5541:
5530:
5527:
5524:
5515:
5512:
5506:
5501:
5498:
5472:
5452:
5449:
5446:
5443:
5434:If a function
5431:
5428:
5427:
5426:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5331:
5328:
5322:
5319:
5316:
5313:
5301:is defined as
5284:
5279:
5266:
5265:
5254:
5248:
5245:
5239:
5236:
5233:
5230:
5223:
5220:
5217:
5212:
5206:
5203:
5197:
5194:
5191:
5188:
5178:
5173:
5167:
5160:
5157:
5154:
5150:
5145:
5142:
5139:
5136:
5130:
5127:
5121:
5118:
5115:
5112:
5088:
5084:
5081:
5061:
5058:
5055:
5035:
5032:
5029:
5006:
5003:
5000:
4980:
4969:
4968:
4957:
4954:
4951:
4948:
4945:
4942:
4939:
4936:
4930:
4927:
4922:
4919:
4913:
4907:
4904:
4899:
4896:
4889:
4886:
4883:
4880:
4877:
4874:
4871:
4868:
4862:
4859:
4854:
4851:
4845:
4839:
4836:
4831:
4828:
4811:
4810:
4799:
4795:
4788:
4785:
4781:
4776:
4773:
4767:
4764:
4760:
4754:
4748:
4745:
4740:
4731:
4728:
4722:
4718:
4704:. Then, using
4693:
4673:
4653:
4650:
4647:
4644:
4641:
4638:
4618:
4615:
4612:
4609:
4606:
4603:
4583:
4580:
4577:
4574:
4571:
4568:
4548:
4545:
4542:
4539:
4536:
4533:
4522:
4521:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4488:
4483:
4479:
4476:
4473:
4470:
4464:
4461:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4408:
4405:
4387:
4386:
4375:
4372:
4369:
4366:
4362:
4359:
4353:
4350:
4344:
4341:
4338:
4335:
4332:
4323:
4320:
4314:
4309:
4303:
4300:
4294:
4291:
4288:
4285:
4282:
4254:
4225:
4210:Main article:
4207:
4204:
4202:
4199:
4195:
4194:
4180:
4177:
4174:
4170:
4165:
4159:
4156:
4153:
4149:
4144:
4137:
4134:
4131:
4125:
4121:
4118:
4111:
4105:
4102:
4099:
4094:
4091:
4088:
4082:
4077:
4074:
4071:
4068:
4065:
4042:
4039:
4019:
3999:
3985:
3984:
3973:
3970:
3967:
3964:
3959:
3955:
3951:
3945:
3941:
3938:
3934:
3930:
3927:
3924:
3919:
3915:
3911:
3908:
3905:
3902:
3897:
3893:
3878:
3877:
3858:
3854:
3851:
3848:
3842:
3838:
3834:
3827:
3824:
3821:
3816:
3813:
3810:
3804:
3798:
3795:
3792:
3788:
3784:
3781:
3778:
3776:
3774:
3771:
3768:
3763:
3759:
3755:
3754:
3748:
3744:
3741:
3738:
3732:
3728:
3721:
3718:
3715:
3710:
3707:
3704:
3698:
3692:
3689:
3685:
3681:
3678:
3675:
3673:
3671:
3668:
3665:
3660:
3656:
3652:
3651:
3625:
3610:
3609:
3598:
3591:
3588:
3585:
3579:
3575:
3572:
3565:
3559:
3556:
3553:
3548:
3545:
3542:
3536:
3531:
3528:
3525:
3522:
3519:
3507:and therefore
3505:
3504:
3493:
3487:
3484:
3481:
3476:
3473:
3470:
3464:
3461:
3455:
3452:
3449:
3444:
3441:
3438:
3432:
3429:
3426:
3421:
3418:
3411:
3408:
3405:
3400:
3397:
3394:
3391:
3387:
3380:
3377:
3374:
3369:
3366:
3363:
3357:
3354:
3351:
3345:
3342:
3339:
3333:
3329:
3326:
3318:
3314:
3307:
3304:
3301:
3297:
3292:
3289:
3286:
3283:
3252:
3249:
3246:
3235:
3234:
3223:
3220:
3217:
3214:
3211:
3208:
3201:
3198:
3195:
3189:
3185:
3182:
3172:
3168:
3163:
3156:
3153:
3150:
3146:
3141:
3138:
3135:
3128:
3125:
3122:
3117:
3114:
3111:
3101:
3097:
3092:
3085:
3082:
3079:
3075:
3050:
3028:
3024:
3001:
2997:
2976:
2973:
2968:
2964:
2960:
2955:
2951:
2923:
2920:
2917:
2897:
2894:
2891:
2888:
2885:
2865:
2845:
2842:
2839:
2836:
2825:
2824:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2790:
2786:
2782:
2779:
2776:
2773:
2770:
2767:
2762:
2758:
2754:
2751:
2726:
2722:
2718:
2715:
2710:
2706:
2678:
2658:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2597:
2594:
2591:
2579:
2576:
2563:
2543:
2519:
2515:
2491:
2487:
2464:
2460:
2439:
2436:
2433:
2430:
2419:
2418:
2406:
2386:
2383:
2380:
2358:
2354:
2330:
2327:
2324:
2321:
2311:
2299:
2279:
2276:
2273:
2268:
2264:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2234:
2230:
2209:
2189:
2167:
2163:
2140:
2136:
2111:
2091:
2088:
2085:
2082:
2058:
2038:
2035:
2032:
2029:
2002:
1990:
1987:
1974:
1971:
1968:
1965:
1945:
1942:
1939:
1936:
1916:
1913:
1910:
1907:
1896:
1895:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1855:
1851:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1821:
1817:
1813:
1810:
1807:
1804:
1781:
1778:
1775:
1755:
1735:
1732:
1729:
1724:
1720:
1699:
1696:
1693:
1688:
1684:
1660:
1639:
1627:
1624:
1611:
1608:
1605:
1602:
1582:
1579:
1576:
1573:
1553:
1550:
1547:
1544:
1533:
1532:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1495:
1491:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1461:
1457:
1453:
1450:
1447:
1444:
1421:
1418:
1415:
1395:
1375:
1372:
1369:
1364:
1360:
1339:
1336:
1333:
1328:
1324:
1305:
1302:
1301:
1300:
1289:
1284:
1280:
1277:
1274:
1269:
1264:
1256:
1253:
1250:
1247:
1242:
1238:
1232:
1229:
1226:
1222:
1207:
1206:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1166:
1162:
1156:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1119:
1116:
1113:
1110:
1105:
1101:
1095:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1034:
1031:
1028:
1023:
1019:
998:
995:
992:
987:
983:
971:
970:
959:
954:
950:
947:
944:
939:
935:
928:
925:
922:
919:
914:
910:
883:
880:
873:
869:
866:
863:
842:
838:
835:
824:
823:
812:
807:
802:
797:
794:
787:
784:
777:
771:
767:
760:
757:
754:
751:
746:
742:
715:
712:
709:
704:
700:
679:
676:
673:
669:
665:
661:
657:
653:
649:
646:
643:
640:
636:
612:
581:
561:
558:
555:
552:
549:
546:
543:
523:
520:
517:
514:
494:
491:
488:
485:
465:
462:
459:
456:
436:
433:
430:
419:
418:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
318:
296:
292:
269:
265:
253:
252:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
194:
190:
160:
156:
133:
129:
106:
76:
64:
61:
57:Israel Gohberg
31:, named after
9:
6:
4:
3:
2:
7436:
7425:
7424:David Hilbert
7422:
7420:
7417:
7415:
7412:
7410:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7366:
7364:
7349:
7348:
7339:
7338:
7335:
7329:
7326:
7324:
7321:
7319:
7316:
7314:
7311:
7309:
7306:
7304:
7301:
7299:
7296:
7294:
7291:
7289:
7286:
7284:
7281:
7279:
7276:
7274:
7271:
7269:
7266:
7264:
7261:
7259:
7256:
7254:
7251:
7249:
7246:
7244:
7241:
7239:
7236:
7234:
7231:
7229:
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7179:
7176:
7174:
7171:
7169:
7166:
7164:
7161:
7159:
7156:
7154:
7151:
7149:
7146:
7144:
7141:
7140:
7137:
7133:
7126:
7121:
7119:
7114:
7112:
7107:
7106:
7103:
7095:
7091:
7087:
7083:
7078:
7073:
7068:
7064:
7058:
7054:
7049:
7046:
7042:
7038:
7034:
7030:
7026:
7025:
7019:
7014:
7009:
7005:
6999:
6995:
6990:
6985:
6979:
6975:
6971:
6967:
6963:
6958:
6954:
6950:
6946:
6942:
6941:
6936:
6935:Lax, Peter D.
6932:
6928:
6924:
6920:
6916:
6912:
6908:
6903:
6898:
6894:
6890:
6885:
6881:
6877:
6873:
6869:
6865:
6861:
6856:
6851:
6848:(7): 073706,
6847:
6843:
6839:
6834:
6830:
6826:
6822:
6818:
6814:
6810:
6805:
6800:
6797:(8): 083505,
6796:
6792:
6791:
6785:
6780:
6775:
6770:
6766:
6759:
6754:
6749:
6745:
6740:
6736:
6732:
6731:
6726:
6721:
6718:
6714:
6713:
6708:
6704:
6700:
6696:
6692:
6688:
6684:
6680:
6676:
6672:
6668:
6667:
6661:
6657:
6653:
6649:
6645:
6641:
6637:
6633:
6629:
6628:
6622:
6618:
6614:
6610:
6606:
6602:
6598:
6594:
6590:
6589:
6584:
6580:
6576:
6572:
6568:
6564:
6560:
6556:
6552:
6547:
6542:
6538:
6534:
6533:
6528:
6524:
6519:
6514:
6510:
6504:
6500:
6496:
6492:
6488:
6483:
6478:
6474:
6470:
6466:
6462:
6457:
6452:
6449:(1): 69–124,
6448:
6444:
6443:
6437:
6434:
6430:
6429:
6424:
6419:
6415:
6411:
6407:
6403:
6402:
6396:
6391:
6386:
6382:
6378:
6377:
6372:
6367:
6363:
6357:
6353:
6348:
6347:
6335:
6330:
6323:
6318:
6311:
6306:
6299:
6294:
6287:
6282:
6275:
6270:
6263:
6258:
6251:
6246:
6239:
6234:
6227:
6226:Bitsadze 2001
6222:
6215:
6210:
6203:
6198:
6194:
6184:
6181:
6179:
6176:
6175:
6169:
6167:
6163:
6158:
6156:
6152:
6148:
6144:
6140:
6136:
6131:
6129:
6125:
6120:
6118:
6114:
6110:
6106:
6101:
6098:
6094:
6090:
6086:
6081:
6079:
6078:Gakhov (2001)
6075:
6074:Cauchy kernel
6071:
6067:
6063:
6059:
6055:
6049:
6047:
6043:
6032:
6030:
6026:
6018:
6014:
6010:
6008:
6004:
6003:
5999:
5994:
5990:
5986:
5984:
5980:
5979:
5975:
5971:
5967:
5965:
5961:
5957:
5956:
5952:
5948:
5944:
5940:
5936:
5932:
5928:
5926:
5922:
5921:
5920:
5912:
5910:
5906:
5902:
5897:
5877:
5854:
5825:
5819:
5816:
5810:
5781:
5775:
5772:
5766:
5746:
5721:
5699:
5676:
5668:
5654:
5648:
5645:
5639:
5636:
5633:
5621:
5615:
5612:
5606:
5603:
5591:
5580:
5567:
5566:
5565:
5551:
5528:
5525:
5522:
5510:
5499:
5486:
5485:
5484:
5470:
5447:
5441:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5383:
5380:
5377:
5374:
5371:
5368:
5362:
5356:
5353:
5350:
5347:
5344:
5341:
5335:
5326:
5320:
5317:
5314:
5311:
5304:
5303:
5302:
5300:
5299:wedge product
5282:
5252:
5243:
5237:
5234:
5231:
5228:
5221:
5218:
5215:
5201:
5195:
5192:
5186:
5176:
5165:
5158:
5155:
5152:
5148:
5143:
5140:
5137:
5125:
5119:
5116:
5110:
5103:
5102:
5101:
5082:
5079:
5053:
5033:
5027:
5018:
5004:
5001:
4998:
4978:
4955:
4949:
4946:
4943:
4937:
4934:
4928:
4920:
4911:
4905:
4897:
4887:
4881:
4878:
4875:
4869:
4866:
4860:
4852:
4843:
4837:
4829:
4816:
4815:
4814:
4797:
4793:
4786:
4774:
4771:
4765:
4752:
4746:
4743:
4738:
4726:
4707:
4706:
4705:
4691:
4671:
4648:
4645:
4642:
4636:
4613:
4610:
4607:
4601:
4578:
4575:
4572:
4566:
4543:
4540:
4537:
4531:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4486:
4481:
4477:
4474:
4471:
4468:
4462:
4459:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4430:
4429:
4428:
4426:
4422:
4392:
4373:
4370:
4367:
4364:
4360:
4348:
4342:
4339:
4333:
4330:
4318:
4298:
4292:
4289:
4283:
4270:
4269:
4268:
4266:
4252:
4242:
4239:
4223:
4213:
4198:
4178:
4175:
4172:
4168:
4163:
4157:
4154:
4151:
4147:
4142:
4135:
4132:
4129:
4123:
4119:
4116:
4109:
4103:
4100:
4097:
4092:
4089:
4086:
4080:
4075:
4069:
4063:
4056:
4055:
4054:
4040:
4037:
4017:
3997:
3988:
3971:
3965:
3957:
3953:
3949:
3943:
3939:
3936:
3932:
3925:
3917:
3913:
3909:
3903:
3895:
3891:
3883:
3882:
3881:
3856:
3852:
3849:
3846:
3840:
3836:
3832:
3825:
3822:
3819:
3814:
3811:
3808:
3796:
3793:
3790:
3786:
3779:
3777:
3769:
3761:
3757:
3746:
3742:
3739:
3736:
3730:
3726:
3719:
3716:
3713:
3708:
3705:
3702:
3690:
3687:
3683:
3676:
3674:
3666:
3658:
3654:
3642:
3641:
3640:
3637:
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3596:
3589:
3586:
3583:
3577:
3573:
3570:
3563:
3557:
3554:
3551:
3546:
3543:
3540:
3534:
3529:
3523:
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3509:
3508:
3491:
3485:
3482:
3479:
3474:
3471:
3468:
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3453:
3450:
3447:
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3436:
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3427:
3424:
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3416:
3409:
3406:
3403:
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3112:
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3090:
3083:
3080:
3077:
3073:
3064:
3063:
3062:
3026:
3022:
2999:
2995:
2974:
2971:
2966:
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2935:
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2889:
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2808:
2805:
2802:
2796:
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2777:
2774:
2768:
2760:
2756:
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2749:
2742:
2741:
2740:
2724:
2720:
2716:
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2704:
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2690:
2676:
2656:
2633:
2630:
2627:
2624:
2618:
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2592:
2589:
2575:
2561:
2533:
2517:
2513:
2489:
2485:
2462:
2458:
2434:
2428:
2378:
2356:
2352:
2344:
2341:tends to the
2325:
2319:
2312:
2274:
2266:
2262:
2255:
2249:
2246:
2240:
2232:
2228:
2187:
2165:
2161:
2138:
2134:
2125:
2124:
2123:
2086:
2080:
2072:
2033:
2027:
2018:
2016:
1986:
1969:
1963:
1940:
1934:
1911:
1905:
1882:
1876:
1870:
1867:
1861:
1853:
1849:
1842:
1836:
1833:
1827:
1819:
1815:
1808:
1802:
1795:
1794:
1793:
1776:
1773:
1730:
1722:
1718:
1694:
1686:
1682:
1674:
1623:
1606:
1600:
1577:
1571:
1548:
1542:
1516:
1510:
1507:
1501:
1493:
1489:
1482:
1476:
1473:
1467:
1459:
1455:
1448:
1442:
1435:
1434:
1433:
1416:
1413:
1370:
1362:
1358:
1334:
1326:
1322:
1311:
1287:
1275:
1267:
1262:
1254:
1248:
1240:
1236:
1224:
1212:
1211:
1210:
1193:
1187:
1181:
1178:
1172:
1164:
1160:
1154:
1147:
1141:
1138:
1135:
1129:
1123:
1117:
1111:
1103:
1099:
1093:
1086:
1080:
1077:
1074:
1068:
1062:
1052:
1051:
1050:
1048:
1029:
1021:
1017:
993:
985:
981:
957:
945:
937:
933:
926:
920:
912:
908:
900:
899:
898:
878:
871:
867:
864:
861:
836:
833:
810:
800:
795:
792:
782:
775:
769:
765:
758:
752:
744:
740:
732:
731:
730:
729:
710:
702:
698:
674:
671:
663:
655:
647:
644:
638:
626:
602:
597:
595:
559:
556:
553:
550:
547:
544:
541:
518:
512:
489:
483:
460:
454:
431:
428:
405:
399:
393:
390:
384:
378:
372:
366:
363:
357:
351:
345:
339:
332:
331:
330:
294:
290:
267:
239:
233:
227:
224:
221:
215:
209:
206:
200:
192:
188:
180:
179:
178:
176:
158:
131:
118:
104:
94:
90:
67:Suppose that
60:
58:
54:
50:
46:
45:complex plane
42:
38:
37:David Hilbert
34:
30:
26:
19:
7345:
7282:
7213:Riemann form
7085:
7081:
7071:
7052:
7028:
7022:
7012:
6993:
6983:
6965:
6961:
6944:
6938:
6892:
6889:Nonlinearity
6888:
6845:
6841:
6794:
6788:
6778:
6768:
6764:
6747:
6743:
6728:
6710:
6703:Gakhov, F.D.
6670:
6664:
6631:
6625:
6592:
6586:
6546:math/9201261
6536:
6530:
6527:Deift, Percy
6517:
6494:
6481:
6446:
6440:
6426:
6405:
6399:
6380:
6374:
6351:
6329:
6317:
6305:
6293:
6281:
6269:
6257:
6245:
6233:
6221:
6209:
6197:
6159:
6155:Toda lattice
6132:
6121:
6109:self-adjoint
6102:
6097:KdV equation
6089:KdV equation
6082:
6050:
6038:
6022:
5998:Dyson (1976)
5992:
5974:Deift (2000)
5949:problem for
5943:Fokas (2002)
5918:
5898:
5692:in a region
5691:
5543:
5433:
5267:
5019:
4971:As such, if
4970:
4812:
4523:
4394:
4391:DBAR problem
4390:
4388:
4215:
4212:DBAR problem
4206:DBAR problem
4196:
3989:
3986:
3879:
3638:
3612:which has a
3611:
3506:
3236:
2936:
2826:
2696:of equation
2691:
2581:
2420:
2019:
2014:
1992:
1897:
1629:
1534:
1313:
1208:
972:
825:
625:circle group
598:
420:
254:
66:
59:and others.
28:
22:
7238:Riemann sum
7055:. Martino.
6968:: 693–696,
6634:(1): 1–39,
6262:Pandey 1996
6238:Pandey 1996
6143:meromorphic
6070:Zhou (1989)
5983:probability
3880:therefore,
3616:at contour
2649:. Assuming
2071:holomorphic
25:mathematics
7363:Categories
6838:Teschl, G.
6750:(3): 14–18
6456:1611.03697
6344:References
6322:Vekua 2014
6274:Noble 1958
6058:Its (1982)
6027:(see e.g.
3614:branch cut
1308:See also:
854:, one has
53:Mark Krein
47:. Several
6927:119171871
6902:1401.2419
6855:0705.0346
6804:0907.5571
6735:EMS Press
6717:EMS Press
6705:(2001) ,
6695:118343085
6656:118630271
6617:122511904
6433:EMS Press
6408:: 39–90,
6107:) is not
5947:monodromy
5881:¯
5829:¯
5785:¯
5725:¯
5672:¯
5658:¯
5625:¯
5595:¯
5586:∂
5578:∂
5514:¯
5505:∂
5497:∂
5411:η
5405:ξ
5393:−
5384:η
5375:−
5372:ξ
5363:∧
5357:η
5345:ξ
5330:¯
5327:ζ
5318:∧
5315:ζ
5247:¯
5244:ζ
5235:∧
5232:ζ
5219:−
5216:ζ
5205:¯
5202:ζ
5193:ζ
5166:∬
5156:π
5129:¯
5060:∞
5057:→
5031:→
5002:∈
4926:∂
4918:∂
4903:∂
4895:∂
4858:∂
4850:∂
4844:−
4835:∂
4827:∂
4784:∂
4780:∂
4763:∂
4759:∂
4730:¯
4721:∂
4717:∂
4407:¯
4404:∂
4368:∈
4352:¯
4322:¯
4313:∂
4302:¯
4281:∂
4155:−
4133:π
4120:
4090:−
4038:−
3958:−
3940:
3918:−
3850:
3841:−
3823:π
3812:
3797:π
3791:−
3762:−
3740:
3717:π
3706:
3691:π
3624:Σ
3587:π
3574:
3544:−
3472:−
3463:
3451:π
3440:
3428:ζ
3420:ζ
3407:−
3396:−
3390:−
3386:∫
3376:π
3365:
3353:ζ
3341:−
3338:ζ
3328:
3317:Σ
3313:∫
3303:π
3285:
3251:Σ
3248:∈
3219:
3210:ζ
3197:−
3194:ζ
3184:
3171:−
3167:Σ
3162:∫
3152:π
3140:−
3137:ζ
3124:−
3121:ζ
3113:
3096:Σ
3091:∫
3081:π
3049:Σ
3027:−
2967:−
2959:−
2922:∞
2919:→
2893:→
2887:
2856:tends to
2809:
2789:−
2781:
2753:
2725:−
2694:logarithm
2625:−
2616:Σ
2542:Σ
2490:−
2405:Σ
2385:∞
2382:→
2298:Σ
2267:−
2208:Σ
2166:−
2110:Σ
2069:, find a
2057:Σ
2001:Σ
1964:γ
1935:β
1906:α
1871:γ
1854:−
1837:β
1803:α
1780:Σ
1777:∈
1754:Σ
1723:−
1659:Σ
1638:Σ
1601:γ
1572:β
1543:α
1511:γ
1494:−
1477:β
1443:α
1420:Σ
1417:∈
1394:Σ
1363:−
1283:¯
1241:−
1231:∞
1228:→
1165:−
1136:−
1047:unit disk
1022:−
953:¯
913:−
882:¯
837:∈
806:¯
793:−
786:¯
745:−
648:∈
611:Σ
580:Σ
435:Σ
432:∈
364:−
317:Σ
264:Σ
159:−
155:Σ
128:Σ
75:Σ
7384:Solitons
7347:Category
6829:17284652
6581:(1976),
6571:12699956
6493:(2000),
6172:See also
6105:Lax pair
6044:and the
5712:, where
5483:, then
5020:In case
4245:complex
4236:is some
4216:Suppose
2941:is that
2582:Suppose
1792:one has
1432:one has
447:, where
97:complex
7033:Bibcode
6970:Bibcode
6907:Bibcode
6880:2579238
6860:Bibcode
6809:Bibcode
6675:Bibcode
6636:Bibcode
6597:Bibcode
6563:2946540
6461:Bibcode
6288:, §7.5.
6276:, §4.2.
6264:, §2.2.
4421:problem
4243:of the
3639:Check:
3267:), the
2220:, then
897:and so
623:is the
599:By the
95:in the
43:in the
7074:, SIAM
7059:
7000:
6986:: 1–77
6925:
6878:
6827:
6693:
6654:
6615:
6569:
6561:
6505:
6358:
4241:domain
2987:where
2827:Since
2532:-sense
1898:where
1535:where
309:along
89:smooth
6923:S2CID
6897:arXiv
6876:S2CID
6850:arXiv
6825:S2CID
6799:arXiv
6761:(PDF)
6691:S2CID
6652:S2CID
6613:S2CID
6567:S2CID
6559:JSTOR
6541:arXiv
6451:arXiv
6189:Notes
6111:, by
6076:(see
4524:with
4265:plane
3237:when
175:index
117:plane
87:is a
7057:ISBN
6998:ISBN
6984:IMRP
6503:ISBN
6356:ISBN
6068:and
6060:and
6025:PDEs
5929:The
5867:and
5803:and
5759:and
5072:and
4684:and
4629:and
4393:(or
4030:and
3014:and
2608:and
2477:and
2153:and
1956:and
1710:and
1593:and
1350:and
1009:and
826:For
572:and
505:and
421:for
35:and
7090:doi
7041:doi
6949:doi
6915:doi
6868:doi
6817:doi
6683:doi
6671:147
6644:doi
6632:230
6605:doi
6551:doi
6537:137
6469:doi
6447:216
6410:doi
6385:doi
6031:).
6017:WKB
5958:B.
5923:A.
5544:in
5046:as
4117:log
3937:log
3847:log
3809:log
3737:log
3703:log
3571:log
3460:log
3437:log
3362:log
3325:log
3282:log
3216:log
3181:log
3110:log
2908:as
2884:log
2806:log
2778:log
2750:log
2371:as
2126:If
1221:lim
596:.
23:In
7365::
7086:20
7084:,
7039:,
7029:37
7027:,
6966:38
6964:,
6945:36
6943:,
6921:,
6913:,
6905:,
6893:28
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6846:53
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6795:46
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6769:50
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6406:37
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6381:12
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6373:,
6168:.
6130:.
6000:).
5976:).
5962:,
5911:.
5896:.
5083::=
4594:,
4559:,
3972:2.
3636:.
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2812:2.
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7110:v
7097:.
7092::
7076:.
7065:.
7043::
7035::
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6988:.
6977:.
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6956:.
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6862::
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6738:.
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6364:.
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6228:.
6204:.
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5993:B
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4438:M
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4340:z
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4290:z
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3667:0
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3410:z
3404:1
3399:z
3393:1
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193:+
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132:+
105:z
20:.
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