298:
77:
2731:
Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have
2452:, that is, infinitely differentiable. The converse is not true for real functions; in fact, in a certain sense, the real analytic functions are sparse compared to all real infinitely differentiable functions. For the complex numbers, the converse does hold, and in fact any function differentiable
2588:
A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function Æ has an
2415:
2869:
on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function
796:
2910:). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up in 2 or more complex dimensions:
2928:
for single-valued functions consist of arbitrary (connected) open sets. In several complex variables, however, only some connected open sets are domains of holomorphy. The characterization of domains of holomorphy leads to the notion of
2739:, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by
1098:
2031:
1667:
2299:
2291:
609:
2488:
1241:
2800:
1273:
2526:
2083:
2696:
1799:
2570:
2202:
2117:
845:
1696:
1580:
1551:
2228:
328:
2698:). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see
1870:
601:
1524:* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from
1916:
3016:
1385:
is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is
1177:
967:
929:
501:
466:
435:
404:
2532:
with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of
1130:
882:
2706:
2248:
2160:
2140:
1936:
1890:
1841:
1819:
1771:
1748:
1728:
1607:
1376:
1356:
1336:
1316:
1296:
1200:
1150:
902:
564:
537:
1941:
2963:
2044:
2656:
975:
1420:
in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own
2736:
321:
3260:
2652:
Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.
1389:
i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.
2824:
1612:
3286:
3127:
3051:
314:
179:
3339:
2705:
The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that
2410:{\displaystyle \sup _{x\in K}\left|{\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)\right|\leq C^{|\alpha |+1}\alpha !}
2043:
For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the
2898:
at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.
41:
This article is about both real and complex analytic functions. For analytic functions in complex analysis specifically, see
2919:
3313:
3252:
2253:
2943:
146:
2968:
2699:
1586:
1509:
functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
184:
174:
791:{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-x_{0}\right)^{n}=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})^{2}+\cdots }
3354:
3308:
2050:
In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let
2437:
of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose
2465:
2442:
1205:
238:
2953:
2745:
1246:
2667:
As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or
2497:
2053:
2670:
372:, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.
2958:
1473:
407:
1776:
3340:
Solver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. Ivanov
2828:
2539:
2434:
1450:
350:
2165:
2088:
2906:
One can define analytic functions in several variables by means of power series in those variables (see
2891:
1179:
804:
254:
56:
46:
3303:
297:
229:
1672:
1556:
1527:
2948:
2925:
2835:
of the complex plane) is not true in general; the function of the example above gives an example for
2719:
2594:
2427:
2037:
1428:
1386:
848:
504:
438:
264:
199:
141:
42:
2988:
2626:
2533:
1402:
151:
113:
31:
2207:
2832:
2806:
2590:
302:
209:
1849:
573:
468:, since every differentiable function has at least a tangent line at every point, which is its
3322:
3282:
3256:
3186:
3133:
3123:
3047:
1513:
852:
354:
224:
108:
3270:
3176:
3043:
3036:
2710:
2646:
2573:
2529:
1895:
1467:
1421:
472:
of order 1. So just having a polynomial expansion at singular points is not enough, and the
284:
279:
269:
245:
160:
131:
122:
98:
68:
2994:
1155:
945:
907:
479:
444:
413:
382:
3274:
2930:
2598:
2449:
1478:
1106:
935:
858:
508:
369:
204:
169:
35:
3204:
3244:
2915:
2456:
on an open set is analytic on that set (see "analyticity and differentiability" below).
2233:
2145:
2125:
1921:
1875:
1826:
1804:
1756:
1733:
1713:
1592:
1500:
1496:
1483:
1458:
1361:
1341:
1321:
1301:
1281:
1185:
1135:
887:
549:
522:
259:
194:
189:
84:
3325:
17:
3348:
2707:
any complex function differentiable (in the complex sense) in an open set is analytic
2577:
939:
473:
469:
376:
219:
214:
103:
2907:
2895:
2581:
1093:{\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}
357:
3164:
1844:
342:
274:
93:
2874:) defined in the paragraph above is a counterexample, as it is not defined for
1298:
defined on some subset of the real line is said to be real analytic at a point
441:. It is important to note that it is a neighborhood and not just at some point
2827:). This statement for real analytic functions (with open ball meaning an open
2438:
1409:
3190:
3181:
2842: = 0 and a ball of radius exceeding 1, since the power series
3330:
3137:
2914:
Zero sets of complex analytic functions in more than one variable are never
1506:
1454:
567:
76:
3117:
2866:
2609:
2460:
544:
1431:
is analytic. Any Taylor series for this function converges not only for
2026:{\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!}
1662:{\displaystyle f\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})}
503:
to be considered an analytic function. As a counterexample see the
2655:
These statements imply that while analytic functions do have more
1503:
is not everywhere analytic because it is not differentiable at 0.
2677:
1625:
1253:
1212:
2805:
Also, if a complex analytic function is defined in an open
2637:, then Æ is identically zero on the connected component of
476:
must also converge to the function on points adjacent to
1491:
Typical examples of functions that are not analytic are
1182:. The set of all real analytic functions on a given set
2601:
containing the accumulation point. In other words, if (
3119:
A guide to distribution theory and
Fourier transforms
2997:
2748:
2673:
2542:
2500:
2468:
2302:
2256:
2236:
2210:
2168:
2148:
2128:
2091:
2056:
2036:
Complex analytic functions are exactly equivalent to
1944:
1924:
1898:
1878:
1852:
1829:
1807:
1779:
1759:
1736:
1716:
1675:
1615:
1595:
1559:
1530:
1364:
1344:
1324:
1304:
1284:
1249:
1208:
1188:
1158:
1138:
1109:
978:
948:
910:
890:
861:
807:
612:
576:
552:
525:
482:
447:
416:
385:
3104:
is also used in the literature do denote analyticity
2286:{\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}
1499:
function when defined on the set of real numbers or
3035:
3010:
2794:
2732:more structure than their real-line counterparts.
2690:
2564:
2520:
2482:
2409:
2285:
2242:
2222:
2196:
2154:
2134:
2111:
2077:
2025:
1930:
1910:
1884:
1864:
1835:
1813:
1793:
1765:
1742:
1722:
1690:
1661:
1601:
1574:
1545:
1370:
1350:
1330:
1310:
1290:
1267:
1235:
1194:
1171:
1144:
1124:
1092:
961:
923:
896:
876:
839:
790:
595:
558:
531:
495:
460:
429:
398:
2991:as well in a (possibly smaller) neighborhood of
2304:
3165:"A characterization of real analytic functions"
2040:, and are thus much more easily characterized.
1470:(at least in some range of the complex plane):
2659:than polynomials, they are still quite rigid.
934:Alternatively, a real analytic function is an
2894:is 1 because the complexified function has a
2483:{\displaystyle \Omega \subseteq \mathbb {C} }
1461:are analytic on any open set of their domain.
1442:(as in the definition) but for all values of
1236:{\displaystyle {\mathcal {C}}^{\,\omega }(D)}
322:
8:
3205:"Gevrey class - Encyclopedia of Mathematics"
3227:
3150:
2964:Infinite compositions of analytic functions
2882:. This explains why the Taylor series of Æ(
1397:Typical examples of analytic functions are
2795:{\displaystyle f(x)={\frac {1}{x^{2}+1}}.}
1268:{\displaystyle {\mathcal {C}}^{\,\omega }}
375:A function is analytic if and only if its
329:
315:
51:
3180:
3002:
2996:
2774:
2764:
2747:
2702:. In fact there are many such functions.
2682:
2676:
2675:
2672:
2547:
2541:
2521:{\displaystyle u:\Omega \to \mathbb {C} }
2514:
2513:
2499:
2476:
2475:
2467:
2388:
2380:
2379:
2349:
2331:
2324:
2307:
2301:
2277:
2269:
2265:
2264:
2255:
2235:
2209:
2179:
2167:
2147:
2127:
2105:
2104:
2090:
2078:{\displaystyle U\subset \mathbb {R} ^{n}}
2069:
2065:
2064:
2055:
2005:
1975:
1957:
1950:
1943:
1923:
1897:
1877:
1851:
1828:
1806:
1787:
1786:
1778:
1758:
1753:There is a complex analytic extension of
1735:
1715:
1707:The following conditions are equivalent:
1682:
1678:
1677:
1674:
1650:
1646:
1645:
1635:
1630:
1624:
1623:
1614:
1594:
1566:
1562:
1561:
1558:
1537:
1533:
1532:
1529:
1363:
1343:
1323:
1303:
1283:
1259:
1258:
1252:
1251:
1248:
1218:
1217:
1211:
1210:
1207:
1187:
1163:
1157:
1137:
1108:
1084:
1074:
1041:
1022:
1015:
1009:
998:
977:
953:
947:
915:
909:
889:
860:
825:
812:
806:
776:
766:
747:
731:
712:
699:
686:
675:
653:
643:
632:
611:
581:
575:
551:
524:
487:
481:
452:
446:
421:
415:
390:
384:
2691:{\displaystyle {\mathcal {C}}^{\infty }}
1589:, and in particular any smooth function
3026:
2980:
2902:Analytic functions of several variables
244:
237:
159:
121:
83:
67:
2823:is convergent in the whole open ball (
2727:Real versus complex analytic functions
3077:if its derivative exists not only at
2831:of the real line rather than an open
1794:{\displaystyle G\subset \mathbb {C} }
45:. For analytic functions in SQL, see
7:
3255:11 (2nd ed.). Springer-Verlag.
3096:if it is analytic at every point in
2565:{\displaystyle A_{\infty }(\Omega )}
3279:A Primer of Real Analytic Functions
3249:Functions of One Complex Variable I
2865:Any real analytic function on some
2597:, then Æ is zero everywhere on the
2430:of analytic functions are analytic.
2197:{\displaystyle f\in C^{\infty }(U)}
2112:{\displaystyle f:U\to \mathbb {R} }
3038:Complex Variables and Applications
2825:holomorphic functions are analytic
2683:
2556:
2548:
2507:
2469:
2342:
2328:
2180:
2045:FourierâBrosâIagolnitzer transform
1636:
1416:, any terms of degree larger than
1010:
936:infinitely differentiable function
840:{\displaystyle a_{0},a_{1},\dots }
644:
406:converges to the function in some
25:
3034:Churchill; Brown; Verhey (1948).
2663:Analyticity and differentiability
3169:Proceedings of the Japan Academy
2816:, its power series expansion at
2612:of distinct numbers such that Æ(
2421:Properties of analytic functions
2250:such that for every multi-index
1730:is real analytic on an open set
1691:{\displaystyle \mathbb {R} ^{n}}
1575:{\displaystyle \mathbb {R} ^{2}}
1546:{\displaystyle \mathbb {R} ^{2}}
296:
75:
2494:(Ω) of all analytic functions
2441:is nowhere zero. (See also the
1918:and every non-negative integer
27:Type of function in mathematics
3116:Strichartz, Robert S. (1994).
2890:| > 1, i.e., the
2758:
2752:
2559:
2553:
2510:
2389:
2381:
2364:
2358:
2191:
2185:
2101:
1990:
1984:
1656:
1641:
1230:
1224:
1119:
1113:
1081:
1061:
1047:
1034:
1029:
1023:
988:
982:
871:
865:
773:
753:
737:
718:
622:
616:
1:
3253:Graduate Texts in Mathematics
3092:. It is analytic in a region
1703:Alternative characterizations
1587:non-analytic smooth functions
1412:: if a polynomial has degree
1275:if the domain is understood.
368:. Functions of each type are
3281:(2nd ed.). BirkhÀuser.
3163:Komatsu, Hikosaburo (1960).
2969:Non-analytic smooth function
2700:non-analytic smooth function
2576:analytic functions with the
2223:{\displaystyle K\subseteq U}
3309:Encyclopedia of Mathematics
2920:Hartogs's extension theorem
1609:with compact support, i.e.
1318:if there is a neighborhood
353:that is locally given by a
3371:
2443:Lagrange inversion theorem
2293:the following bound holds
1938:the following bound holds
1865:{\displaystyle K\subset D}
801:in which the coefficients
596:{\displaystyle x_{0}\in D}
366:complex analytic functions
40:
29:
3122:. Boca Raton: CRC Press.
2448:Any analytic function is
1383:complex analytic function
847:are real numbers and the
370:infinitely differentiable
239:Geometric function theory
185:Cauchy's integral formula
175:Cauchy's integral theorem
3085:in some neighborhood of
3062:of the complex variable
2944:CauchyâRiemann equations
2918:. This can be proved by
2621:) = 0 for all
2426:The sums, products, and
2230:there exists a constant
2085:be an open set, and let
1872:there exists a constant
1843:is smooth and for every
1669:, cannot be analytic on
1474:hypergeometric functions
147:CauchyâRiemann equations
30:Not to be confused with
3228:Krantz & Parks 2002
3151:Krantz & Parks 2002
3042:. McGraw-Hill. p.
2959:Quasi-analytic function
2645:. This is known as the
1451:trigonometric functions
362:real analytic functions
132:Complex-valued function
3209:encyclopediaofmath.org
3182:10.3792/pja/1195524081
3012:
2796:
2692:
2566:
2522:
2484:
2411:
2287:
2244:
2224:
2204:and for every compact
2198:
2156:
2136:
2113:
2079:
2027:
1932:
1912:
1911:{\displaystyle x\in K}
1886:
1866:
1837:
1815:
1795:
1767:
1744:
1724:
1692:
1663:
1603:
1576:
1547:
1372:
1352:
1332:
1312:
1292:
1269:
1237:
1196:
1173:
1146:
1126:
1094:
1014:
963:
925:
898:
878:
841:
792:
648:
597:
560:
533:
497:
462:
431:
400:
303:Mathematics portal
18:Real analytic function
3013:
3011:{\displaystyle x_{0}}
2926:Domains of holomorphy
2892:radius of convergence
2797:
2693:
2567:
2523:
2485:
2412:
2288:
2245:
2225:
2199:
2157:
2137:
2114:
2080:
2038:holomorphic functions
2028:
1933:
1913:
1887:
1867:
1838:
1816:
1796:
1768:
1745:
1725:
1693:
1664:
1604:
1577:
1548:
1373:
1353:
1333:
1313:
1293:
1270:
1238:
1197:
1174:
1172:{\displaystyle x_{0}}
1152:in a neighborhood of
1147:
1127:
1095:
994:
964:
962:{\displaystyle x_{0}}
926:
924:{\displaystyle x_{0}}
904:in a neighborhood of
899:
879:
842:
793:
628:
598:
561:
534:
519:Formally, a function
498:
496:{\displaystyle x_{0}}
463:
461:{\displaystyle x_{0}}
432:
430:{\displaystyle x_{0}}
401:
399:{\displaystyle x_{0}}
255:Augustin-Louis Cauchy
57:Mathematical analysis
47:Window function (SQL)
2995:
2954:PaleyâWiener theorem
2949:Holomorphic function
2746:
2720:holomorphic function
2671:
2540:
2498:
2466:
2300:
2254:
2234:
2208:
2166:
2146:
2142:is real analytic on
2126:
2089:
2054:
1942:
1922:
1896:
1892:such that for every
1876:
1850:
1827:
1805:
1777:
1757:
1734:
1714:
1673:
1613:
1593:
1557:
1528:
1429:exponential function
1403:elementary functions
1381:The definition of a
1362:
1342:
1322:
1302:
1282:
1247:
1206:
1202:is often denoted by
1186:
1156:
1136:
1125:{\displaystyle f(x)}
1107:
976:
946:
908:
888:
877:{\displaystyle f(x)}
859:
805:
610:
574:
550:
523:
505:Weierstrass function
480:
445:
414:
383:
265:Carl Friedrich Gauss
200:Isolated singularity
142:Holomorphic function
43:holomorphic function
3326:"Analytic Function"
3304:"Analytic function"
2989:uniform convergence
2737:Liouville's theorem
2717:is synonymous with
2709:. Consequently, in
2599:connected component
2282:
1640:
360:. There exist both
152:Formal power series
114:Unit complex number
32:analytic expression
3355:Analytic functions
3323:Weisstein, Eric W.
3081:but at each point
3008:
2792:
2688:
2657:degrees of freedom
2625:and this sequence
2591:accumulation point
2562:
2518:
2480:
2407:
2318:
2283:
2263:
2240:
2220:
2194:
2152:
2132:
2109:
2075:
2023:
1928:
1908:
1882:
1862:
1833:
1811:
1791:
1763:
1740:
1720:
1688:
1659:
1622:
1599:
1572:
1543:
1446:(real or complex).
1378:is real analytic.
1368:
1348:
1328:
1308:
1288:
1265:
1233:
1192:
1169:
1142:
1122:
1090:
959:
921:
894:
874:
837:
788:
593:
556:
529:
493:
458:
427:
396:
230:Laplace's equation
210:Argument principle
3262:978-0-387-90328-6
2862:| â„ 1.
2787:
2715:analytic function
2633:in the domain of
2356:
2303:
2243:{\displaystyle C}
2155:{\displaystyle U}
2135:{\displaystyle f}
1982:
1931:{\displaystyle k}
1885:{\displaystyle C}
1836:{\displaystyle f}
1814:{\displaystyle D}
1766:{\displaystyle f}
1743:{\displaystyle D}
1723:{\displaystyle f}
1602:{\displaystyle f}
1514:complex conjugate
1507:Piecewise defined
1468:special functions
1371:{\displaystyle f}
1351:{\displaystyle x}
1331:{\displaystyle D}
1311:{\displaystyle x}
1291:{\displaystyle f}
1195:{\displaystyle D}
1145:{\displaystyle x}
1059:
897:{\displaystyle x}
559:{\displaystyle D}
532:{\displaystyle f}
347:analytic function
339:
338:
225:Harmonic function
137:Analytic function
123:Complex functions
109:Complex conjugate
16:(Redirected from
3362:
3336:
3335:
3317:
3292:
3275:Parks, Harold R.
3266:
3231:
3225:
3219:
3218:
3216:
3215:
3201:
3195:
3194:
3184:
3160:
3154:
3148:
3142:
3141:
3113:
3107:
3106:
3041:
3031:
3019:
3017:
3015:
3014:
3009:
3007:
3006:
2985:
2886:) diverges for |
2857:
2801:
2799:
2798:
2793:
2788:
2786:
2779:
2778:
2765:
2711:complex analysis
2697:
2695:
2694:
2689:
2687:
2686:
2681:
2680:
2647:identity theorem
2571:
2569:
2568:
2563:
2552:
2551:
2534:Morera's theorem
2527:
2525:
2524:
2519:
2517:
2489:
2487:
2486:
2481:
2479:
2416:
2414:
2413:
2408:
2400:
2399:
2392:
2384:
2371:
2367:
2357:
2355:
2354:
2353:
2340:
2336:
2335:
2325:
2317:
2292:
2290:
2289:
2284:
2281:
2276:
2268:
2249:
2247:
2246:
2241:
2229:
2227:
2226:
2221:
2203:
2201:
2200:
2195:
2184:
2183:
2161:
2159:
2158:
2153:
2141:
2139:
2138:
2133:
2118:
2116:
2115:
2110:
2108:
2084:
2082:
2081:
2076:
2074:
2073:
2068:
2032:
2030:
2029:
2024:
2016:
2015:
1997:
1993:
1983:
1981:
1980:
1979:
1966:
1962:
1961:
1951:
1937:
1935:
1934:
1929:
1917:
1915:
1914:
1909:
1891:
1889:
1888:
1883:
1871:
1869:
1868:
1863:
1842:
1840:
1839:
1834:
1820:
1818:
1817:
1812:
1800:
1798:
1797:
1792:
1790:
1772:
1770:
1769:
1764:
1749:
1747:
1746:
1741:
1729:
1727:
1726:
1721:
1697:
1695:
1694:
1689:
1687:
1686:
1681:
1668:
1666:
1665:
1660:
1655:
1654:
1649:
1639:
1634:
1629:
1628:
1608:
1606:
1605:
1600:
1581:
1579:
1578:
1573:
1571:
1570:
1565:
1552:
1550:
1549:
1544:
1542:
1541:
1536:
1479:Bessel functions
1435:close enough to
1422:Maclaurin series
1377:
1375:
1374:
1369:
1357:
1355:
1354:
1349:
1337:
1335:
1334:
1329:
1317:
1315:
1314:
1309:
1297:
1295:
1294:
1289:
1274:
1272:
1271:
1266:
1264:
1263:
1257:
1256:
1242:
1240:
1239:
1234:
1223:
1222:
1216:
1215:
1201:
1199:
1198:
1193:
1178:
1176:
1175:
1170:
1168:
1167:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1099:
1097:
1096:
1091:
1089:
1088:
1079:
1078:
1060:
1058:
1050:
1046:
1045:
1033:
1032:
1016:
1013:
1008:
968:
966:
965:
960:
958:
957:
930:
928:
927:
922:
920:
919:
903:
901:
900:
895:
883:
881:
880:
875:
846:
844:
843:
838:
830:
829:
817:
816:
797:
795:
794:
789:
781:
780:
771:
770:
752:
751:
736:
735:
717:
716:
704:
703:
691:
690:
685:
681:
680:
679:
658:
657:
647:
642:
602:
600:
599:
594:
586:
585:
565:
563:
562:
557:
538:
536:
535:
530:
502:
500:
499:
494:
492:
491:
467:
465:
464:
459:
457:
456:
436:
434:
433:
428:
426:
425:
405:
403:
402:
397:
395:
394:
331:
324:
317:
301:
300:
285:Karl Weierstrass
280:Bernhard Riemann
270:Jacques Hadamard
99:Imaginary number
79:
69:Complex analysis
63:
61:Complex analysis
52:
21:
3370:
3369:
3365:
3364:
3363:
3361:
3360:
3359:
3345:
3344:
3321:
3320:
3302:
3299:
3289:
3269:
3263:
3245:Conway, John B.
3243:
3240:
3235:
3234:
3226:
3222:
3213:
3211:
3203:
3202:
3198:
3162:
3161:
3157:
3149:
3145:
3130:
3115:
3114:
3110:
3091:
3076:
3054:
3033:
3032:
3028:
3023:
3022:
2998:
2993:
2992:
2986:
2982:
2977:
2940:
2931:pseudoconvexity
2904:
2843:
2841:
2822:
2815:
2809:around a point
2770:
2769:
2744:
2743:
2729:
2674:
2669:
2668:
2665:
2620:
2606:
2543:
2538:
2537:
2496:
2495:
2464:
2463:
2423:
2375:
2345:
2341:
2327:
2326:
2323:
2319:
2298:
2297:
2252:
2251:
2232:
2231:
2206:
2205:
2175:
2164:
2163:
2162:if and only if
2144:
2143:
2124:
2123:
2087:
2086:
2063:
2052:
2051:
2001:
1971:
1967:
1953:
1952:
1949:
1945:
1940:
1939:
1920:
1919:
1894:
1893:
1874:
1873:
1848:
1847:
1825:
1824:
1803:
1802:
1801:which contains
1775:
1774:
1773:to an open set
1755:
1754:
1732:
1731:
1712:
1711:
1705:
1676:
1671:
1670:
1644:
1611:
1610:
1591:
1590:
1560:
1555:
1554:
1531:
1526:
1525:
1501:complex numbers
1484:gamma functions
1459:power functions
1441:
1395:
1360:
1359:
1340:
1339:
1320:
1319:
1300:
1299:
1280:
1279:
1250:
1245:
1244:
1209:
1204:
1203:
1184:
1183:
1159:
1154:
1153:
1134:
1133:
1105:
1104:
1080:
1070:
1051:
1037:
1018:
1017:
974:
973:
949:
944:
943:
911:
906:
905:
886:
885:
857:
856:
821:
808:
803:
802:
772:
762:
743:
727:
708:
695:
671:
664:
660:
659:
649:
608:
607:
603:one can write
577:
572:
571:
548:
547:
521:
520:
517:
509:Fabius function
483:
478:
477:
448:
443:
442:
417:
412:
411:
386:
381:
380:
335:
295:
205:Residue theorem
180:Local primitive
170:Zeros and poles
85:Complex numbers
55:
50:
39:
36:analytic signal
28:
23:
22:
15:
12:
11:
5:
3368:
3366:
3358:
3357:
3347:
3346:
3343:
3342:
3337:
3318:
3298:
3297:External links
3295:
3294:
3293:
3287:
3271:Krantz, Steven
3267:
3261:
3239:
3236:
3233:
3232:
3220:
3196:
3155:
3143:
3128:
3108:
3089:
3074:
3052:
3025:
3024:
3021:
3020:
3005:
3001:
2979:
2978:
2976:
2973:
2972:
2971:
2966:
2961:
2956:
2951:
2946:
2939:
2936:
2935:
2934:
2923:
2903:
2900:
2878: = ±
2858:diverges for |
2839:
2820:
2813:
2803:
2802:
2791:
2785:
2782:
2777:
2773:
2768:
2763:
2760:
2757:
2754:
2751:
2728:
2725:
2685:
2679:
2664:
2661:
2616:
2604:
2586:
2585:
2561:
2558:
2555:
2550:
2546:
2516:
2512:
2509:
2506:
2503:
2478:
2474:
2471:
2457:
2446:
2431:
2422:
2419:
2418:
2417:
2406:
2403:
2398:
2395:
2391:
2387:
2383:
2378:
2374:
2370:
2366:
2363:
2360:
2352:
2348:
2344:
2339:
2334:
2330:
2322:
2316:
2313:
2310:
2306:
2280:
2275:
2272:
2267:
2262:
2259:
2239:
2219:
2216:
2213:
2193:
2190:
2187:
2182:
2178:
2174:
2171:
2151:
2131:
2107:
2103:
2100:
2097:
2094:
2072:
2067:
2062:
2059:
2034:
2033:
2022:
2019:
2014:
2011:
2008:
2004:
2000:
1996:
1992:
1989:
1986:
1978:
1974:
1970:
1965:
1960:
1956:
1948:
1927:
1907:
1904:
1901:
1881:
1861:
1858:
1855:
1832:
1822:
1810:
1789:
1785:
1782:
1762:
1751:
1739:
1719:
1704:
1701:
1700:
1699:
1685:
1680:
1658:
1653:
1648:
1643:
1638:
1633:
1627:
1621:
1618:
1598:
1583:
1569:
1564:
1540:
1535:
1520: →
1510:
1504:
1497:absolute value
1489:
1488:
1487:
1486:
1481:
1476:
1464:
1463:
1462:
1447:
1439:
1425:
1401:The following
1394:
1391:
1367:
1347:
1327:
1307:
1287:
1262:
1255:
1232:
1229:
1226:
1221:
1214:
1191:
1166:
1162:
1141:
1121:
1118:
1115:
1112:
1101:
1100:
1087:
1083:
1077:
1073:
1069:
1066:
1063:
1057:
1054:
1049:
1044:
1040:
1036:
1031:
1028:
1025:
1021:
1012:
1007:
1004:
1001:
997:
993:
990:
987:
984:
981:
969:in its domain
956:
952:
938:such that the
918:
914:
893:
873:
870:
867:
864:
836:
833:
828:
824:
820:
815:
811:
799:
798:
787:
784:
779:
775:
769:
765:
761:
758:
755:
750:
746:
742:
739:
734:
730:
726:
723:
720:
715:
711:
707:
702:
698:
694:
689:
684:
678:
674:
670:
667:
663:
656:
652:
646:
641:
638:
635:
631:
627:
624:
621:
618:
615:
592:
589:
584:
580:
555:
528:
516:
513:
490:
486:
455:
451:
424:
420:
393:
389:
337:
336:
334:
333:
326:
319:
311:
308:
307:
306:
305:
290:
289:
288:
287:
282:
277:
272:
267:
262:
260:Leonhard Euler
257:
249:
248:
242:
241:
235:
234:
233:
232:
227:
222:
217:
212:
207:
202:
197:
195:Laurent series
192:
190:Winding number
187:
182:
177:
172:
164:
163:
157:
156:
155:
154:
149:
144:
139:
134:
126:
125:
119:
118:
117:
116:
111:
106:
101:
96:
88:
87:
81:
80:
72:
71:
65:
64:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3367:
3356:
3353:
3352:
3350:
3341:
3338:
3333:
3332:
3327:
3324:
3319:
3315:
3311:
3310:
3305:
3301:
3300:
3296:
3290:
3288:0-8176-4264-1
3284:
3280:
3276:
3272:
3268:
3264:
3258:
3254:
3250:
3246:
3242:
3241:
3237:
3229:
3224:
3221:
3210:
3206:
3200:
3197:
3192:
3188:
3183:
3178:
3174:
3170:
3166:
3159:
3156:
3153:, p. 15.
3152:
3147:
3144:
3139:
3135:
3131:
3129:0-8493-8273-4
3125:
3121:
3120:
3112:
3109:
3105:
3103:
3099:
3095:
3088:
3084:
3080:
3073:
3069:
3065:
3061:
3055:
3053:0-07-010855-2
3049:
3045:
3040:
3039:
3030:
3027:
3003:
2999:
2990:
2987:This implies
2984:
2981:
2974:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2941:
2937:
2932:
2927:
2924:
2921:
2917:
2913:
2912:
2911:
2909:
2901:
2899:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2868:
2863:
2861:
2855:
2851:
2847:
2838:
2834:
2830:
2826:
2819:
2812:
2808:
2789:
2783:
2780:
2775:
2771:
2766:
2761:
2755:
2749:
2742:
2741:
2740:
2738:
2735:According to
2733:
2726:
2724:
2722:
2721:
2716:
2712:
2708:
2703:
2701:
2662:
2660:
2658:
2653:
2650:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2619:
2615:
2611:
2607:
2600:
2596:
2592:
2583:
2579:
2578:supremum norm
2575:
2544:
2535:
2531:
2530:Fréchet space
2504:
2501:
2493:
2472:
2462:
2458:
2455:
2451:
2447:
2444:
2440:
2436:
2432:
2429:
2425:
2424:
2420:
2404:
2401:
2396:
2393:
2385:
2376:
2372:
2368:
2361:
2350:
2346:
2337:
2332:
2320:
2314:
2311:
2308:
2296:
2295:
2294:
2278:
2273:
2270:
2260:
2257:
2237:
2217:
2214:
2211:
2188:
2176:
2172:
2169:
2149:
2129:
2120:
2098:
2095:
2092:
2070:
2060:
2057:
2048:
2046:
2041:
2039:
2020:
2017:
2012:
2009:
2006:
2002:
1998:
1994:
1987:
1976:
1972:
1968:
1963:
1958:
1954:
1946:
1925:
1905:
1902:
1899:
1879:
1859:
1856:
1853:
1846:
1830:
1823:
1808:
1783:
1780:
1760:
1752:
1737:
1717:
1710:
1709:
1708:
1702:
1683:
1651:
1631:
1619:
1616:
1596:
1588:
1584:
1567:
1538:
1523:
1519:
1515:
1511:
1508:
1505:
1502:
1498:
1494:
1493:
1492:
1485:
1482:
1480:
1477:
1475:
1472:
1471:
1469:
1465:
1460:
1456:
1452:
1448:
1445:
1438:
1434:
1430:
1426:
1423:
1419:
1415:
1411:
1407:
1406:
1404:
1400:
1399:
1398:
1392:
1390:
1388:
1384:
1379:
1365:
1345:
1325:
1305:
1285:
1276:
1260:
1243:, or just by
1227:
1219:
1189:
1181:
1164:
1160:
1139:
1116:
1110:
1103:converges to
1085:
1075:
1071:
1067:
1064:
1055:
1052:
1042:
1038:
1026:
1019:
1005:
1002:
999:
995:
991:
985:
979:
972:
971:
970:
954:
950:
942:at any point
941:
940:Taylor series
937:
932:
916:
912:
891:
868:
862:
854:
850:
834:
831:
826:
822:
818:
813:
809:
785:
782:
777:
767:
763:
759:
756:
748:
744:
740:
732:
728:
724:
721:
713:
709:
705:
700:
696:
692:
687:
682:
676:
672:
668:
665:
661:
654:
650:
639:
636:
633:
629:
625:
619:
613:
606:
605:
604:
590:
587:
582:
578:
569:
553:
546:
542:
541:real analytic
526:
514:
512:
510:
506:
488:
484:
475:
474:Taylor series
471:
470:Taylor series
453:
449:
440:
422:
418:
409:
391:
387:
378:
377:Taylor series
373:
371:
367:
363:
359:
356:
352:
348:
344:
332:
327:
325:
320:
318:
313:
312:
310:
309:
304:
299:
294:
293:
292:
291:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
261:
258:
256:
253:
252:
251:
250:
247:
243:
240:
236:
231:
228:
226:
223:
221:
220:Schwarz lemma
218:
216:
215:Conformal map
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
167:
166:
165:
162:
158:
153:
150:
148:
145:
143:
140:
138:
135:
133:
130:
129:
128:
127:
124:
120:
115:
112:
110:
107:
105:
104:Complex plane
102:
100:
97:
95:
92:
91:
90:
89:
86:
82:
78:
74:
73:
70:
66:
62:
58:
54:
53:
48:
44:
37:
33:
19:
3329:
3307:
3278:
3248:
3223:
3212:. Retrieved
3208:
3199:
3175:(3): 90â93.
3172:
3168:
3158:
3146:
3118:
3111:
3101:
3100:. The term
3097:
3093:
3086:
3082:
3078:
3071:
3067:
3063:
3059:
3057:
3037:
3029:
2983:
2908:power series
2905:
2887:
2883:
2879:
2875:
2871:
2864:
2859:
2853:
2849:
2845:
2836:
2817:
2810:
2804:
2734:
2730:
2718:
2714:
2704:
2666:
2654:
2651:
2642:
2638:
2634:
2630:
2622:
2617:
2613:
2602:
2587:
2582:Banach space
2491:
2453:
2428:compositions
2121:
2049:
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408:neighborhood
374:
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361:
358:power series
346:
340:
161:Basic theory
136:
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3102:holomorphic
3058:A function
2713:, the term
2641:containing
2629:to a point
2593:inside its
1845:compact set
1410:polynomials
1387:holomorphic
1278:A function
570:if for any
515:Definitions
343:mathematics
275:Kiyoshi Oka
94:Real number
3238:References
3214:2020-08-30
2536:. The set
2490:, the set
2439:derivative
2435:reciprocal
1457:, and the
853:convergent
410:for every
355:convergent
3331:MathWorld
3314:EMS Press
3191:0021-4280
3070:at point
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1999:≤
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1620:∈
1516:function
1455:logarithm
1358:on which
1261:ω
1220:ω
1180:pointwise
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645:∞
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588:∈
568:real line
3349:Category
3277:(2002).
3247:(1978).
3138:28890674
3068:analytic
2938:See also
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1393:Examples
545:open set
351:function
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1585:Other
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543:on an
439:domain
379:about
246:People
2975:Notes
2580:is a
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2122:Then
1466:Most
349:is a
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3283:ISBN
3257:ISBN
3187:ISSN
3134:OCLC
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