568:
43:
345:
497:
2575:
2228:
2860:
124:
2066:
1476:
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the
2954:) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful
2223:{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}
1471:
3020:, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a
807:
1011:
964:
1997:
2399:
1819:
917:
1612:
2324:, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see
1918:
3043:
about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.
2470:
1776:
507:
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include
1172:
878:
2708:
2058:
1701:
2829:
with determinant one). Some authors define conformality to include orientation-reversing mappings whose
Jacobians can be written as any scalar times any orthogonal matrix.
2356:
2322:
1739:
375:
2279:
2773:
2434:
2740:
2023:
1558:
1538:
1291:
1267:
2934:. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, as is stated by the
3016:, which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the
2804:
1502:
1340:
1504:
in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are
670:
1649:
2679:
2659:
2616:
2596:
2555:
2533:
2512:
2492:
1313:
1094:
1074:
1054:
1034:
641:
1520:, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on
2806:, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or
2837:
1781:
An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the
3548:
2986:
2978:, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials.
1348:
1180:) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as
686:
2557:
is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset.
485:
368:
3195:
3162:
3028:
471:
1778:
are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below).
64:
1184:, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every
31:
3008:
domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be
361:
226:
86:
3600:
2938:. The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in
3680:
3607:
3082:
2814:
969:
1560:. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are
925:
567:
3590:
3385:
2990:
2947:
1782:
604:
to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a
3595:
3541:
3351:
3035:
expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as
193:
3459:
2939:
1565:
1208:
in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of
231:
221:
3467:
2325:
1926:
3503:
2942:). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of
2916:
2832:
For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible
2363:
57:
51:
1788:
886:
3260:
3149:
2998:
2943:
1193:
1587:
3769:
3475:
2818:
1825:
1505:
285:
3097:
1516:+ 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of
68:
3774:
3534:
2847:
3012:
from its values on the smaller domain. This allows the extension of the definition of functions, such as the
3040:
2935:
1225:
524:
463:
2440:
3712:
3612:
3182:. The Student Mathematical Library. Vol. 9. Providence, Rhode Island: American Mathematical Society.
2631:
1576:
1135:
597:
536:
456:
452:
416:
400:
1746:
1477:
difference quotient must approach the same complex number, regardless of the manner in which we approach
3705:
3700:
3664:
3660:
3585:
3558:
3072:
3047:
3013:
3009:
2963:
1201:
590:
548:
500:
448:
396:
301:
103:
1145:
815:
344:
276:
2684:
2030:
1671:
1663:
is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as
3732:
3637:
3226:
2951:
2833:
2002:
1572:
1237:
1217:
1185:
1119:
605:
556:
512:
480:
311:
246:
188:
3727:
3717:
3670:
3494:
2994:
2843:
2821:. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a
1177:
444:
420:
198:
160:
2339:
2284:
1708:
3749:
3632:
3455:
3395:
3371:
3201:
2244:
408:
349:
256:
3333:
3102:
2958:. The remarkable behavior of holomorphic functions near essential singularities is described by
2745:
2406:
3722:
3685:
3274:
3191:
3158:
3087:
3051:
3017:
2826:
2713:
1517:
1221:
1213:
1205:
1189:
1181:
1104:
532:
475:
436:
271:
183:
155:
17:
2008:
1543:
1523:
1276:
1252:
3675:
3498:
3418:
3319:
3222:
3183:
3067:
2982:
677:
540:
516:
496:
443:. By extension, use of complex analysis also has applications in engineering fields such as
331:
326:
316:
145:
2782:
1540:
can be approximated arbitrarily well by polynomials in some neighborhood of every point in
1480:
1318:
3690:
3627:
3622:
3617:
3405:
3337:
3077:
3021:
3005:
2955:
2863:
2822:
1197:
646:
404:
251:
216:
3343:
3027:
All this refers to complex analysis in one variable. There is also a very rich theory of
1626:
3526:
3445:
3361:
3307:
3290:
2971:
2959:
2924:
2664:
2644:
2601:
2581:
2540:
2518:
2497:
2477:
2332:
1298:
1243:
1209:
1079:
1059:
1039:
1019:
626:
601:
572:
440:
432:
306:
241:
236:
131:
3509:
3479:
3311:
3282:
3763:
3694:
3650:
3251:
Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes.
3205:
3092:
3055:
2975:
2931:
2627:
2618:
maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
2567:
617:
552:
467:
428:
412:
266:
261:
150:
3521:
1566:
Non-analytic smooth function § A smooth function which is nowhere real analytic
612:. Complex functions are generally assumed to have a domain that contains a nonempty
3428:
3246:
3236:
3036:
3032:
1115:
531:, and many more in the 20th century. Complex analysis, in particular the theory of
3345:
A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable
2335:
asserts that the range of an entire function can take only three possible forms:
2967:
2623:
673:
613:
528:
321:
140:
3580:
2920:
1580:
1111:
1466:{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}
1196:
of a point are equal on the intersection of their domain (if the domains are
3278:
3255:
Elementary Theory of
Analytic Functions of One or Several Complex Variables.
3122:
2807:
802:{\displaystyle z=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}
2574:
2985:
that is holomorphic in the entire complex plane must be constant; this is
2859:
123:
3737:
3575:
3570:
1247:
1192:(see next section), and two differentiable functions that are equal in a
609:
2842:
The notion of conformality generalizes in a natural way to maps between
3187:
2331:
Holomorphic functions exhibit some remarkable features. For instance,
582:
544:
539:. In modern times, it has become very popular through a new boost from
424:
1659:
are polynomials, are holomorphic on domains that exclude points where
520:
2858:
2776:
2635:
2573:
566:
508:
3530:
2989:. It can be used to provide a natural and short proof for the
2912:
1114:
to, and therefore, in that sense, it) can be considered as an
36:
27:
Branch of mathematics studying functions of a complex variable
2578:
A rectangular grid (top) and its image under a conformal map
535:, has many physical applications and is also used throughout
3413:
1616:
are holomorphic over the entire complex plane, making them
1583:, extended appropriately to complex arguments as functions
551:. Another important application of complex analysis is in
407:. It is helpful in many branches of mathematics, including
2233:
In terms of the real and imaginary parts of the function,
1216:
complex functions are defined in this way, including the
1006:{\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}
3380:
Methods of the Theory of
Functions of a Complex Variable
2813:
The conformal property may be described in terms of the
1200:). The latter property is the basis of the principle of
959:{\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }
3284:
Functions of a
Complex Variable: Theory and Technique.
2840:
sharply limits the conformal mappings to a few types.
1295:
In the context of complex analysis, the derivative of
2785:
2748:
2742:
is called conformal (or angle-preserving) at a point
2716:
2687:
2667:
2647:
2604:
2584:
2543:
2521:
2500:
2480:
2443:
2409:
2366:
2342:
2287:
2247:
2069:
2033:
2011:
1929:
1828:
1791:
1749:
1711:
1674:
1629:
1590:
1546:
1526:
1483:
1351:
1321:
1301:
1279:
1255:
1176:
Some properties of complex-valued functions (such as
1148:
1082:
1062:
1042:
1022:
972:
928:
889:
818:
689:
649:
629:
2930:
One of the central tools in complex analysis is the
3029:
complex analysis in more than one complex dimension
1512:
th derivative need not imply the existence of the (
474:), complex analysis is particularly concerned with
3522:Wolfram Research's MathWorld Complex Analysis Page
2798:
2767:
2734:
2702:
2673:
2653:
2610:
2590:
2549:
2527:
2506:
2486:
2464:
2428:
2393:
2350:
2316:
2273:
2222:
2052:
2017:
1991:
1912:
1813:
1770:
1733:
1695:
1643:
1606:
1552:
1532:
1496:
1465:
1334:
1307:
1285:
1261:
1166:
1088:
1068:
1048:
1028:
1005:
958:
911:
872:
801:
664:
635:
3342:. (Gauthier-Villars, 1905). English translation,
1992:{\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} }
3231:Complex Variables: Introduction and Applications
2474:In other words, if two distinct complex numbers
1380:
3376:Методы теории функций комплексного переменного.
2394:{\displaystyle \mathbb {C} \setminus \{z_{0}\}}
2241:, this is equivalent to the pair of equations
1814:{\displaystyle f:\mathbb {C} \to \mathbb {C} }
912:{\displaystyle f:\mathbb {C} \to \mathbb {C} }
3542:
3469:The Theory Of Functions Of A Complex Variable
3123:"Industrial Applications of Complex Analysis"
369:
8:
2423:
2410:
2388:
2375:
1607:{\displaystyle \mathbb {C} \to \mathbb {C} }
2836:functions. In three and higher dimensions,
2514:are not in the range of an entire function
1913:{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}
3549:
3535:
3527:
3356:Applied and Computational Complex Analysis
3004:If a function is holomorphic throughout a
376:
362:
98:
3390:Theory of Functions of a Complex Variable
3326:. (Springer, 1995). English translation,
3313:Theory of Functions of a Complex Variable
3269:Theory of Functions of a Complex Variable
3267:(Birkhäuser, 1950). English translation,
3031:in which the analytic properties such as
2790:
2784:
2753:
2747:
2715:
2694:
2690:
2689:
2686:
2666:
2646:
2603:
2583:
2542:
2520:
2499:
2479:
2458:
2457:
2448:
2442:
2417:
2408:
2382:
2368:
2367:
2365:
2344:
2343:
2341:
2308:
2292:
2286:
2265:
2252:
2246:
2197:
2176:
2161:
2156:
2142:
2141:
2132:
2127:
2106:
2085:
2084:
2070:
2068:
2038:
2032:
2010:
1985:
1984:
1928:
1827:
1807:
1806:
1799:
1798:
1790:
1757:
1756:
1748:
1726:
1718:
1710:
1673:
1633:
1628:
1600:
1599:
1592:
1591:
1589:
1571:Most elementary functions, including the
1545:
1525:
1488:
1482:
1451:
1430:
1402:
1394:
1383:
1367:
1350:
1326:
1320:
1300:
1278:
1254:
1155:
1151:
1150:
1147:
1081:
1061:
1041:
1021:
996:
995:
986:
982:
981:
971:
951:
950:
941:
937:
936:
927:
905:
904:
897:
896:
888:
817:
709:
688:
648:
628:
503:, one of the founders of complex analysis
393:theory of functions of a complex variable
87:Learn how and when to remove this message
2962:. Functions that have only poles but no
2775:if it preserves angles between directed
495:
50:This article includes a list of general
3114:
2372:
1099:Similarly, any complex-valued function
555:which examines conformal invariants in
291:
284:
206:
168:
130:
114:
3464:Теория функций комплексной переменной.
3253:(Hermann, 1961). English translation,
3180:Inversion Theory and Conformal Mapping
3157:. McGraw-Hill Education. p. 197.
1016:i.e., into two real-valued functions (
486:functions of several complex variables
466:of a complex variable is equal to its
178:
169:
2974:are the complex-valued equivalent to
2465:{\displaystyle z_{0}\in \mathbb {C} }
623:For any complex function, the values
7:
3466:(Nauka, 1967). English translation,
3339:Cours d'analyse mathématique, tome 2
3304:(Wadsworth & Brooks/Cole, 1990).
1667:. On the other hand, the functions
1269:of the complex plane are said to be
1771:{\displaystyle z\mapsto {\bar {z}}}
883:In other words, a complex function
672:in the range may be separated into
3295:Functions of One Complex Variable.
2203:
2199:
2182:
2178:
2138:
2134:
2081:
2073:
2047:
2012:
1681:
1547:
1527:
1280:
1256:
1204:which allows extending every real
484:. The concept can be extended to
207:
56:it lacks sufficient corresponding
25:
3440:Complex Analysis with Mathematica
3366:Advanced Engineering Mathematics.
1167:{\displaystyle \mathbb {R} ^{2}.}
873:{\displaystyle x,y,u(x,y),v(x,y)}
643:from the domain and their images
2946:among others is applicable (see
2703:{\displaystyle \mathbb {R} ^{n}}
2566:This section is an excerpt from
2053:{\displaystyle z_{0}\in \Omega }
1696:{\displaystyle z\mapsto \Re (z)}
478:of a complex variable, that is,
343:
122:
41:
3681:Least-squares spectral analysis
3608:Fundamental theorem of calculus
3083:List of complex analysis topics
3046:A major application of certain
2638:, but not necessarily lengths.
1508:, whereas the existence of the
1186:differentiable complex function
973:
955:
714:
708:
2991:fundamental theorem of algebra
2948:methods of contour integration
2726:
2147:
2112:
2099:
2090:
1978:
1966:
1957:
1945:
1907:
1895:
1883:
1871:
1862:
1847:
1838:
1832:
1803:
1762:
1753:
1727:
1719:
1715:
1690:
1684:
1678:
1596:
1436:
1423:
1414:
1408:
1387:
1373:
1360:
992:
947:
901:
867:
855:
846:
834:
793:
781:
769:
757:
748:
733:
724:
718:
659:
653:
18:Function of a complex variable
1:
3127:Newton Gateway to Mathematics
608:and the complex numbers as a
391:, traditionally known as the
2351:{\displaystyle \mathbb {C} }
2317:{\displaystyle u_{y}=-v_{x}}
1734:{\displaystyle z\mapsto |z|}
1218:complex exponential function
427:, including the branches of
292:
3504:A Course of Modern Analysis
3423:Theory of Complex Functions
3178:Blair, David (2000-08-17).
2274:{\displaystyle u_{x}=v_{y}}
1623:, while rational functions
1242:Complex functions that are
1222:complex logarithm functions
3791:
3433:Real and Complex Analysis.
2768:{\displaystyle u_{0}\in U}
2598:(bottom). It is seen that
2565:
1235:
29:
3746:
3646:
3565:
3414:http://usf.usfca.edu/vca/
3392:, (Prentice-Hall, 1965).
3151:Real and Complex Analysis
3039:) do not carry over. The
2940:Cauchy's integral formula
2848:semi-Riemannian manifolds
2819:coordinate transformation
2429:{\displaystyle \{z_{0}\}}
1783:Cauchy–Riemann conditions
1506:infinitely differentiable
1056:) of two real variables (
286:Geometric function theory
232:Cauchy's integral formula
222:Cauchy's integral theorem
3489:Visual Complex Functions
3481:The Theory of Functions.
3410:Visual Complex Analysis.
2735:{\displaystyle f:U\to V}
1212:removed. Many basic and
1134:or, alternatively, as a
596:A complex function is a
194:Cauchy–Riemann equations
30:Not to be confused with
3400:Basic Complex Analysis.
3281:, & C. E. Pearson,
3257:(Addison-Wesley, 1963).
3041:Riemann mapping theorem
2964:essential singularities
2936:Cauchy integral theorem
2817:derivative matrix of a
2634:that locally preserves
2326:Looman–Menchoff theorem
2018:{\displaystyle \Omega }
1577:trigonometric functions
1553:{\displaystyle \Omega }
1533:{\displaystyle \Omega }
1286:{\displaystyle \Omega }
1262:{\displaystyle \Omega }
1226:trigonometric functions
919:may be decomposed into
464:differentiable function
179:Complex-valued function
71:more precise citations.
3613:Calculus of variations
3586:Differential equations
3382:). (1951, in Russian).
3148:Rudin, Walter (1987).
3010:analytically continued
2997:of complex numbers is
2993:which states that the
2927:
2800:
2769:
2736:
2704:
2675:
2655:
2619:
2612:
2592:
2551:
2529:
2508:
2488:
2466:
2430:
2395:
2352:
2318:
2275:
2224:
2054:
2019:
1993:
1914:
1815:
1772:
1735:
1697:
1645:
1608:
1554:
1534:
1498:
1467:
1336:
1309:
1287:
1263:
1168:
1136:vector-valued function
1090:
1070:
1050:
1030:
1007:
960:
913:
874:
803:
666:
637:
593:
547:produced by iterating
537:analytic number theory
504:
457:electrical engineering
417:analytic combinatorics
350:Mathematics portal
3706:Representation theory
3665:quaternionic analysis
3661:Hypercomplex analysis
3559:mathematical analysis
3491:. (Birkhäuser, 2012).
3073:Hypercomplex analysis
3014:Riemann zeta function
2862:
2801:
2799:{\displaystyle u_{0}}
2770:
2737:
2705:
2676:
2656:
2613:
2593:
2577:
2552:
2530:
2509:
2489:
2467:
2431:
2396:
2353:
2319:
2276:
2225:
2055:
2020:
1994:
1915:
1816:
1773:
1736:
1698:
1665:meromorphic functions
1646:
1609:
1555:
1535:
1499:
1497:{\displaystyle z_{0}}
1468:
1337:
1335:{\displaystyle z_{0}}
1310:
1288:
1264:
1246:at every point of an
1232:Holomorphic functions
1202:analytic continuation
1169:
1120:real-valued functions
1091:
1071:
1051:
1031:
1008:
961:
914:
880:are all real-valued.
875:
804:
667:
638:
591:geometric progression
570:
549:holomorphic functions
501:Augustin-Louis Cauchy
499:
481:holomorphic functions
397:mathematical analysis
302:Augustin-Louis Cauchy
104:Mathematical analysis
3638:Table of derivatives
3448:& R. Shakarchi,
3435:(McGraw-Hill, 1966).
3287:(McGraw-Hill, 1966).
3243:(McGraw-Hill, 1953).
3098:Riemann–Roch theorem
2999:algebraically closed
2952:isolated singularity
2783:
2746:
2714:
2685:
2665:
2645:
2602:
2582:
2541:
2519:
2498:
2478:
2441:
2407:
2364:
2340:
2285:
2245:
2067:
2031:
2009:
2001:is holomorphic on a
1927:
1826:
1789:
1747:
1709:
1672:
1627:
1588:
1581:polynomial functions
1573:exponential function
1544:
1524:
1481:
1349:
1319:
1299:
1277:
1253:
1238:Holomorphic function
1146:
1080:
1060:
1040:
1020:
970:
926:
887:
816:
687:
665:{\displaystyle f(z)}
647:
627:
557:quantum field theory
543:and the pictures of
525:Gösta Mittag-Leffler
312:Carl Friedrich Gauss
247:Isolated singularity
189:Holomorphic function
3718:Continuous function
3671:Functional analysis
3508:(Cambridge, 1902).
3425:. (Springer, 1990).
3386:Markushevich, A. I.
3330:. (Springer, 2005).
2987:Liouville's theorem
2838:Liouville's theorem
2681:be open subsets of
2641:More formally, let
1644:{\displaystyle p/q}
421:applied mathematics
395:, is the branch of
199:Formal power series
161:Unit complex number
3750:Mathematics portal
3633:Lists of integrals
3452:(Princeton, 2003).
3442:(Cambridge, 2006).
3316:(Cambridge, 1893).
3302:Complex Variables.
3265:Funktionentheorie.
3233:(Cambridge, 2003).
3129:. October 30, 2019
2928:
2796:
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2015:
1989:
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1046:
1026:
1003:
956:
909:
870:
799:
662:
633:
594:
533:conformal mappings
505:
476:analytic functions
409:algebraic geometry
399:that investigates
277:Laplace's equation
257:Argument principle
3757:
3756:
3723:Special functions
3686:Harmonic analysis
3476:Titchmarsh, E. C.
3456:Sveshnikov, A. G.
3450:Complex Analysis.
3374:& B. Shabat,
3324:Funktionentheorie
3297:(Springer, 1973).
3271:(Chelsea, 1954).
3197:978-0-8218-2636-2
3164:978-0-07-054234-1
3088:Monodromy theorem
3052:quantum mechanics
3018:natural logarithm
2864:Color wheel graph
2674:{\displaystyle V}
2654:{\displaystyle U}
2611:{\displaystyle f}
2591:{\displaystyle f}
2550:{\displaystyle f}
2528:{\displaystyle f}
2507:{\displaystyle w}
2487:{\displaystyle z}
2210:
2189:
2169:
2154:
2150:
2130:
2126:
2097:
2093:
1765:
1458:
1379:
1342:is defined to be
1308:{\displaystyle f}
1206:analytic function
1182:differentiability
1089:{\displaystyle y}
1069:{\displaystyle x}
1049:{\displaystyle v}
1029:{\displaystyle u}
712:
636:{\displaystyle z}
563:Complex functions
437:quantum mechanics
386:
385:
272:Harmonic function
184:Analytic function
170:Complex functions
156:Complex conjugate
97:
96:
89:
32:Complexity theory
16:(Redirected from
3782:
3770:Complex analysis
3676:Fourier analysis
3656:Complex analysis
3557:Major topics in
3551:
3544:
3537:
3528:
3495:Whittaker, E. T.
3412:(Oxford, 1997).
3402:(Freeman, 1973).
3328:Complex Analysis
3322:& R. Busam,
3261:Carathéodory, C.
3241:Complex Analysis
3210:
3209:
3188:10.1090/stml/009
3175:
3169:
3168:
3156:
3145:
3139:
3138:
3136:
3134:
3119:
3068:Complex geometry
2983:bounded function
2960:Picard's theorem
2950:). A "pole" (or
2909:
2908:
2906:
2905:
2896:
2893:
2866:of the function
2834:complex analytic
2805:
2803:
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2397:
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2359:
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2354:
2349:
2347:
2333:Picard's theorem
2323:
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2016:
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1133:
1109:
1103:on an arbitrary
1102:
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713:
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671:
669:
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663:
642:
640:
639:
634:
588:
580:
541:complex dynamics
470:(that is, it is
423:, as well as in
389:Complex analysis
378:
371:
364:
348:
347:
332:Karl Weierstrass
327:Bernhard Riemann
317:Jacques Hadamard
146:Imaginary number
126:
116:Complex analysis
110:
108:Complex analysis
99:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
3790:
3789:
3785:
3784:
3783:
3781:
3780:
3779:
3775:Complex numbers
3760:
3759:
3758:
3753:
3742:
3691:P-adic analysis
3642:
3628:Matrix calculus
3623:Tensor calculus
3618:Vector calculus
3581:Differentiation
3561:
3555:
3518:
3484:(Oxford, 1932).
3398:& Hoffman,
3348:. (Ginn, 1916).
3223:Ablowitz, M. J.
3219:
3214:
3213:
3198:
3177:
3176:
3172:
3165:
3154:
3147:
3146:
3142:
3132:
3130:
3121:
3120:
3116:
3111:
3103:Runge's theorem
3078:Vector calculus
3064:
3022:Riemann surface
2956:residue theorem
2915:represents the
2911:
2897:
2894:
2879:
2878:
2876:
2867:
2857:
2852:
2851:
2823:rotation matrix
2786:
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2005:
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1234:
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884:
814:
813:
711: and
685:
684:
645:
644:
625:
624:
602:complex numbers
586:
581:of a discrete (
576:
565:
494:
405:complex numbers
382:
342:
252:Residue theorem
227:Local primitive
217:Zeros and poles
132:Complex numbers
102:
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
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3740:
3735:
3730:
3725:
3720:
3715:
3709:
3708:
3703:
3701:Measure theory
3698:
3695:P-adic numbers
3688:
3683:
3678:
3673:
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3643:
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3553:
3546:
3539:
3531:
3525:
3524:
3517:
3516:External links
3514:
3513:
3512:
3510:3rd ed. (1920)
3492:
3485:
3473:
3460:A. N. Tikhonov
3453:
3443:
3436:
3426:
3416:
3403:
3393:
3383:
3372:Lavrentyev, M.
3369:
3368:(Wiley, 1962).
3359:
3349:
3331:
3317:
3305:
3298:
3288:
3275:Carrier, G. F.
3272:
3258:
3244:
3234:
3218:
3215:
3212:
3211:
3196:
3170:
3163:
3140:
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3107:
3106:
3105:
3100:
3095:
3090:
3085:
3080:
3075:
3070:
3063:
3060:
3056:wave functions
3048:complex spaces
2972:Laurent series
2856:
2853:
2793:
2789:
2764:
2761:
2756:
2752:
2731:
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2075:
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2014:
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1692:
1689:
1686:
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1677:
1640:
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1602:
1598:
1594:
1564:analytic; see
1549:
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1473:
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1378:
1375:
1370:
1366:
1362:
1358:
1355:
1329:
1325:
1304:
1282:
1271:holomorphic on
1258:
1244:differentiable
1236:Main article:
1233:
1230:
1163:
1158:
1153:
1085:
1065:
1045:
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989:
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695:
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658:
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632:
564:
561:
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441:twistor theory
433:thermodynamics
384:
383:
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366:
358:
355:
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307:Leonhard Euler
304:
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269:
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249:
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242:Laurent series
239:
237:Winding number
234:
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3:
2:
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3692:
3689:
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3684:
3682:
3679:
3677:
3674:
3672:
3669:
3666:
3662:
3659:
3657:
3654:
3652:
3651:Real analysis
3649:
3648:
3645:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
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3611:
3609:
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3602:
3599:
3597:
3594:
3592:
3589:
3588:
3587:
3584:
3582:
3579:
3577:
3573:
3572:
3568:
3567:
3564:
3560:
3552:
3547:
3545:
3540:
3538:
3533:
3532:
3529:
3523:
3520:
3519:
3515:
3511:
3507:
3505:
3500:
3496:
3493:
3490:
3486:
3483:
3482:
3477:
3474:
3471:
3470:
3465:
3461:
3457:
3454:
3451:
3447:
3444:
3441:
3438:Shaw, W. T.,
3437:
3434:
3430:
3427:
3424:
3420:
3417:
3415:
3411:
3407:
3404:
3401:
3397:
3394:
3391:
3387:
3384:
3381:
3377:
3373:
3370:
3367:
3363:
3360:
3357:
3353:
3350:
3347:
3346:
3341:
3340:
3335:
3332:
3329:
3325:
3321:
3318:
3315:
3314:
3309:
3306:
3303:
3299:
3296:
3292:
3291:Conway, J. B.
3289:
3286:
3285:
3280:
3276:
3273:
3270:
3266:
3262:
3259:
3256:
3252:
3248:
3245:
3242:
3238:
3235:
3232:
3228:
3224:
3221:
3220:
3216:
3207:
3203:
3199:
3193:
3189:
3185:
3181:
3174:
3171:
3166:
3160:
3153:
3152:
3144:
3141:
3128:
3124:
3118:
3115:
3108:
3104:
3101:
3099:
3096:
3094:
3093:Real analysis
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3071:
3069:
3066:
3065:
3061:
3059:
3057:
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3049:
3044:
3042:
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3025:
3023:
3019:
3015:
3011:
3007:
3002:
3000:
2996:
2992:
2988:
2984:
2979:
2977:
2976:Taylor series
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2933:
2932:line integral
2926:
2922:
2918:
2914:
2904:
2900:
2891:
2887:
2883:
2874:
2870:
2865:
2861:
2855:Major results
2854:
2849:
2845:
2841:
2839:
2835:
2830:
2828:
2824:
2820:
2816:
2811:
2809:
2791:
2787:
2778:
2762:
2759:
2754:
2750:
2729:
2723:
2720:
2717:
2710:. A function
2695:
2668:
2648:
2639:
2637:
2633:
2629:
2628:conformal map
2625:
2605:
2585:
2576:
2569:
2568:Conformal map
2561:Conformal map
2560:
2558:
2544:
2522:
2501:
2481:
2454:
2449:
2445:
2418:
2414:
2383:
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2327:
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2185:
2172:
2166:
2163:
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2118:
2115:
2107:
2103:
2087:
2076:
2063:
2062:
2061:
2044:
2039:
2035:
2027:then for all
2004:
1981:
1975:
1972:
1969:
1963:
1960:
1954:
1951:
1948:
1942:
1939:
1936:
1933:
1930:
1904:
1901:
1898:
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1889:
1886:
1880:
1877:
1874:
1868:
1865:
1859:
1856:
1853:
1850:
1844:
1841:
1835:
1829:
1821:, defined by
1795:
1792:
1784:
1779:
1759:
1750:
1723:
1712:
1687:
1675:
1666:
1662:
1658:
1654:
1638:
1634:
1630:
1622:
1619:
1582:
1578:
1574:
1569:
1567:
1563:
1519:
1515:
1511:
1507:
1489:
1485:
1460:
1452:
1448:
1444:
1441:
1431:
1427:
1420:
1417:
1411:
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1391:
1384:
1376:
1368:
1364:
1356:
1353:
1345:
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1323:
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1272:
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1239:
1231:
1229:
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1191:
1187:
1183:
1179:
1174:
1161:
1156:
1137:
1131:
1127:
1121:
1117:
1113:
1106:
1097:
1083:
1063:
1043:
1023:
1000:
987:
977:
974:
942:
932:
929:
922:
921:
920:
893:
890:
881:
864:
861:
858:
852:
849:
843:
840:
837:
831:
828:
825:
822:
819:
796:
790:
787:
784:
778:
775:
772:
766:
763:
760:
754:
751:
745:
742:
739:
736:
730:
727:
721:
715:
705:
702:
699:
696:
693:
690:
683:
682:
681:
679:
675:
656:
650:
630:
621:
619:
618:complex plane
615:
611:
607:
603:
599:
592:
589:, similar to
584:
579:
574:
569:
562:
560:
558:
554:
553:string theory
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
502:
498:
491:
489:
487:
483:
482:
477:
473:
469:
468:Taylor series
465:
460:
458:
454:
450:
446:
442:
438:
434:
430:
429:hydrodynamics
426:
422:
418:
414:
413:number theory
410:
406:
402:
398:
394:
390:
379:
374:
372:
367:
365:
360:
359:
357:
356:
351:
346:
341:
340:
339:
338:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
299:
298:
297:
294:
290:
287:
283:
278:
275:
273:
270:
268:
267:Schwarz lemma
265:
263:
262:Conformal map
260:
258:
255:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
214:
213:
212:
209:
205:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
176:
175:
174:
171:
167:
162:
159:
157:
154:
152:
151:Complex plane
149:
147:
144:
142:
139:
138:
137:
136:
133:
129:
125:
121:
120:
117:
113:
109:
105:
101:
100:
91:
88:
80:
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
3748:
3655:
3569:
3502:
3499:G. N. Watson
3488:
3487:Wegert, E.,
3480:
3472:(MIR, 1978).
3468:
3463:
3449:
3439:
3432:
3422:
3409:
3399:
3389:
3379:
3375:
3365:
3362:Kreyszig, E.
3355:
3344:
3338:
3327:
3323:
3312:
3301:
3300:Fisher, S.,
3294:
3283:
3268:
3264:
3254:
3250:
3240:
3230:
3179:
3173:
3150:
3143:
3133:November 20,
3131:. Retrieved
3126:
3117:
3045:
3037:conformality
3033:power series
3026:
3003:
2980:
2929:
2902:
2898:
2889:
2885:
2881:
2872:
2868:
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1116:ordered pair
1098:
1015:
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595:
577:
506:
479:
461:
392:
388:
387:
208:Basic theory
115:
107:
83:
74:
55:
3576:Integration
3419:Remmert, R.
3406:Needham, T.
3352:Henrici, P.
3334:Goursat, E.
3320:Freitag, E.
3308:Forsyth, A.
3237:Ahlfors, L.
3227:A. S. Fokas
2968:meromorphic
2966:are called
2624:mathematics
2129:where
1518:analyticity
1248:open subset
614:open subset
585:) variable
573:exponential
529:Weierstrass
322:Kiyoshi Oka
141:Real number
69:introducing
3764:Categories
3601:stochastic
3358:(Wiley).
3247:Cartan, H.
3109:References
2925:magnitude.
2921:brightness
2844:Riemannian
2827:orthogonal
1579:, and all
1210:curve arcs
1178:continuity
1112:isomorphic
453:mechanical
77:March 2021
52:references
3713:Functions
3446:Stein, E.
3429:Rudin, W.
3206:118752074
3006:connected
2808:curvature
2760:∈
2727:→
2455:∈
2436:for some
2373:∖
2302:−
2204:∂
2200:∂
2183:∂
2179:∂
2148:¯
2139:∂
2135:∂
2091:¯
2082:∂
2074:∂
2048:Ω
2045:∈
2013:Ω
1982:∈
1804:→
1763:¯
1754:↦
1716:↦
1682:ℜ
1679:↦
1621:functions
1597:→
1548:Ω
1528:Ω
1445:−
1418:−
1388:→
1281:Ω
1257:Ω
1198:connected
993:→
948:→
902:→
678:imaginary
575:function
449:aerospace
401:functions
3738:Infinity
3591:ordinary
3571:Calculus
3279:M. Krook
3062:See also
2944:residues
2917:argument
2815:Jacobian
2779:through
2632:function
1651:, where
1357:′
1190:analytic
610:codomain
598:function
545:fractals
472:analytic
3596:partial
3396:Marsden
3217:Sources
2907:
2901:+ 2 + 2
2877:
1562:nowhere
1214:special
1118:of two
680:parts:
616:of the
583:integer
517:Riemann
492:History
445:nuclear
425:physics
65:improve
3733:Series
3497:&
3458:&
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2777:curves
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2003:region
1922:where
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1618:entire
1575:, the
1224:, and
812:where
606:domain
521:Cauchy
439:, and
419:, and
293:People
54:, but
3728:Limit
3202:S2CID
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2995:field
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2630:is a
2537:then
1142:into
1138:from
1128:, Im
600:from
513:Gauss
509:Euler
462:As a
3192:ISBN
3159:ISBN
3135:2023
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2661:and
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