287:
66:
2720:
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
2441:, 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
2577:
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
2404:
2858:
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
785:
2899:). 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:
2917:
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
2728:, 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
1087:
2020:
1656:
2288:
2280:
598:
2477:
1230:
2789:
1262:
2515:
2072:
2685:
1788:
2559:
2191:
2106:
834:
1685:
1569:
1540:
2217:
2687:). (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
1859:
590:
1513:* 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
1905:
3005:
1374:
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
1166:
956:
918:
490:
455:
424:
393:
2521:
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
1119:
871:
2237:
2149:
2129:
1925:
1879:
1830:
1808:
1760:
1737:
1717:
1596:
1365:
1345:
1325:
1305:
1285:
1189:
1139:
891:
553:
526:
1930:
964:
1409:
in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own
317:
2695:
2641:
Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component.
1378:
i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.
2952:
1601:
2645:
2033:
2725:
3328:
310:
2694:
The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that
2399:{\displaystyle \sup _{x\in K}\left|{\frac {\partial ^{\alpha }f}{\partial x^{\alpha }}}(x)\right|\leq C^{|\alpha |+1}\alpha !}
2032:
For the case of an analytic function with several variables (see below), the real analyticity can be characterized using the
3249:
2887:
at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.
30:
This article is about both real and complex analytic functions. For analytic functions in complex analysis specifically, see
2813:
2242:
1498:
functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
780:{\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 }
2039:
In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let
3275:
3116:
3040:
2426:
of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose
303:
168:
2454:
1194:
2734:
1235:
2908:
2656:
As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or
2486:
2042:
2659:
3302:
3241:
361:, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.
2957:
2932:
2688:
1765:
1575:
173:
163:
135:
3329:
Solver for all zeros of a complex analytic function that lie within a rectangular region by Ivan B. Ivanov
2528:
3343:
3297:
2154:
2077:
2895:
One can define analytic functions in several variables by means of power series in those variables (see
793:
3292:
2431:
1661:
1545:
1516:
2824:
of the complex plane) is not true in general; the function of the example above gives an example for
227:
120:
111:
2942:
2196:
2947:
1462:
396:
149:
1838:
562:
2817:
2423:
1439:
457:, since every differentiable function has at least a tangent line at every point, which is its
339:
2880:
1168:
243:
218:
45:
35:
3032:
3025:
1884:
461:
of order 1. So just having a polynomial expansion at singular points is not enough, and the
2983:
2937:
2914:
2708:
2583:
2416:
2026:
1417:
1375:
1144:
934:
896:
837:
493:
468:
433:
427:
402:
371:
253:
234:
188:
130:
31:
1095:
847:
8:
3193:
2977:
2615:
2522:
1391:
140:
102:
20:
2821:
2795:
2579:
2445:
on an open set is analytic on that set (see "analyticity and differentiability" below).
2222:
2134:
2114:
1910:
1864:
1815:
1793:
1745:
1722:
1702:
1581:
1350:
1330:
1310:
1290:
1270:
1174:
1124:
876:
538:
511:
291:
198:
3314:
3311:
3271:
3245:
3175:
3122:
3112:
3036:
2696:
any complex function differentiable (in the complex sense) in an open set is analytic
1502:
841:
343:
286:
213:
97:
3259:
3165:
2699:
2635:
2562:
1456:
1410:
1082:{\displaystyle T(x)=\sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}
273:
268:
258:
87:
57:
3153:
2518:
3263:
2919:
2587:
2438:
1467:
924:
497:
358:
193:
158:
24:
3233:
2904:
2863:) defined in the paragraph above is a counterexample, as it is not defined for
1489:
1485:
1472:
1447:
1287:
defined on some subset of the real line is said to be real analytic at a point
430:. It is important to note that it is a neighborhood and not just at some point
248:
183:
178:
73:
2816:). This statement for real analytic functions (with open ball meaning an open
3337:
3179:
3170:
2831: = 0 and a ball of radius exceeding 1, since the power series
2566:
928:
462:
458:
365:
208:
203:
92:
3126:
2903:
Zero sets of complex analytic functions in more than one variable are never
2896:
2884:
2570:
346:
3106:
1833:
331:
263:
82:
1420:
is analytic. Any Taylor series for this function converges not only for
2427:
1398:
2015:{\displaystyle \left|{\frac {d^{k}f}{dx^{k}}}(x)\right|\leq C^{k+1}k!}
3319:
1495:
1443:
556:
2855:
2598:
2449:
1651:{\displaystyle f\in {\mathcal {C}}_{0}^{\infty }(\mathbb {R} ^{n})}
533:
492:
to be considered an analytic function. As a counterexample see the
2644:
These statements imply that while analytic functions do have more
1492:
is not everywhere analytic because it is not differentiable at 0.
65:
2794:
Also, if a complex analytic function is defined in an open
2626:, then Æ is identically zero on the connected component of
3309:
2890:
2715:
465:
must also converge to the function on points adjacent to
3133:
1480:
Typical examples of functions that are not analytic are
1171:. The set of all real analytic functions on a given set
2590:
containing the accumulation point. In other words, if (
3210:
3108:
A guide to distribution theory and
Fourier transforms
3104:
2986:
2737:
2662:
2531:
2489:
2457:
2291:
2245:
2225:
2199:
2157:
2137:
2117:
2080:
2045:
2025:
Complex analytic functions are exactly equivalent to
1933:
1913:
1887:
1867:
1841:
1818:
1796:
1768:
1748:
1725:
1705:
1664:
1604:
1584:
1548:
1519:
1353:
1333:
1313:
1293:
1273:
1238:
1197:
1177:
1147:
1127:
1098:
967:
937:
899:
879:
850:
796:
601:
565:
541:
514:
471:
436:
405:
374:
3093:
is also used in the literature do denote analyticity
3022:
2651:
2275:{\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}
1488:
function when defined on the set of real numbers or
2409:
3024:
2999:
2783:
2721:more structure than their real-line counterparts.
2679:
2553:
2509:
2471:
2398:
2274:
2231:
2211:
2185:
2143:
2123:
2100:
2066:
2014:
1919:
1899:
1873:
1853:
1824:
1802:
1782:
1754:
1731:
1711:
1679:
1650:
1590:
1563:
1534:
1359:
1339:
1319:
1299:
1279:
1256:
1224:
1183:
1160:
1133:
1113:
1081:
950:
912:
885:
865:
828:
779:
584:
547:
520:
484:
449:
418:
387:
3335:
2980:as well in a (possibly smaller) neighborhood of
2293:
1691:
3154:"A characterization of real analytic functions"
2029:, and are thus much more easily characterized.
1459:(at least in some range of the complex plane):
2648:than polynomials, they are still quite rigid.
923:Alternatively, a real analytic function is an
2883:is 1 because the complexified function has a
2472:{\displaystyle \Omega \subseteq \mathbb {C} }
1450:are analytic on any open set of their domain.
1431:(as in the definition) but for all values of
1225:{\displaystyle {\mathcal {C}}^{\,\omega }(D)}
311:
3194:"Gevrey class - Encyclopedia of Mathematics"
3258:
3216:
3139:
2953:Infinite compositions of analytic functions
2871:. This explains why the Taylor series of Æ(
1386:Typical examples of analytic functions are
2784:{\displaystyle f(x)={\frac {1}{x^{2}+1}}.}
1257:{\displaystyle {\mathcal {C}}^{\,\omega }}
364:A function is analytic if and only if its
318:
304:
3169:
2691:. In fact there are many such functions.
2510:{\displaystyle u:\Omega \to \mathbb {C} }
2503:
2465:
2254:
2094:
2067:{\displaystyle U\subset \mathbb {R} ^{n}}
2054:
1776:
1742:There is a complex analytic extension of
1696:The following conditions are equivalent:
1667:
1635:
1551:
1522:
1248:
1207:
2680:{\displaystyle {\mathcal {C}}^{\infty }}
1578:, and in particular any smooth function
3151:
2891:Analytic functions of several variables
3336:
3232:
2812:is convergent in the whole open ball (
2716:Real versus complex analytic functions
3310:
3066:if its derivative exists not only at
2820:of the real line rather than an open
1783:{\displaystyle G\subset \mathbb {C} }
34:. For analytic functions in SQL, see
3244:11 (2nd ed.). Springer-Verlag.
3085:if it is analytic at every point in
2554:{\displaystyle A_{\infty }(\Omega )}
3268:A Primer of Real Analytic Functions
3238:Functions of One Complex Variable I
2854:Any real analytic function on some
2586:, then Æ is zero everywhere on the
2419:of analytic functions are analytic.
2186:{\displaystyle f\in C^{\infty }(U)}
2101:{\displaystyle f:U\to \mathbb {R} }
13:
3027:Complex Variables and Applications
2814:holomorphic functions are analytic
2672:
2666:
2545:
2537:
2496:
2458:
2331:
2317:
2169:
2034:FourierâBrosâIagolnitzer transform
1625:
1614:
1405:, any terms of degree larger than
1242:
1201:
999:
925:infinitely differentiable function
829:{\displaystyle a_{0},a_{1},\dots }
633:
395:converges to the function in some
14:
3355:
3285:
3023:Churchill; Brown; Verhey (1948).
2652:Analyticity and differentiability
3158:Proceedings of the Japan Academy
2805:, its power series expansion at
2601:of distinct numbers such that Æ(
2410:Properties of analytic functions
2239:such that for every multi-index
1719:is real analytic on an open set
1680:{\displaystyle \mathbb {R} ^{n}}
1564:{\displaystyle \mathbb {R} ^{2}}
1535:{\displaystyle \mathbb {R} ^{2}}
285:
64:
2483:(Ω) of all analytic functions
2430:is nowhere zero. (See also the
1907:and every non-negative integer
16:Type of function in mathematics
3186:
3145:
3105:Strichartz, Robert S. (1994).
3098:
3016:
2970:
2879:| > 1, i.e., the
2747:
2741:
2548:
2542:
2499:
2378:
2370:
2353:
2347:
2180:
2174:
2090:
1979:
1973:
1645:
1630:
1219:
1213:
1108:
1102:
1070:
1050:
1036:
1023:
1018:
1012:
977:
971:
860:
854:
762:
742:
726:
707:
611:
605:
503:
1:
3242:Graduate Texts in Mathematics
3226:
3081:. It is analytic in a region
1692:Alternative characterizations
1576:non-analytic smooth functions
1401:: if a polynomial has degree
1264:if the domain is understood.
357:. Functions of each type are
3270:(2nd ed.). BirkhÀuser.
3152:Komatsu, Hikosaburo (1960).
2958:Non-analytic smooth function
2689:non-analytic smooth function
2565:analytic functions with the
2212:{\displaystyle K\subseteq U}
7:
3298:Encyclopedia of Mathematics
2926:
2909:Hartogs's extension theorem
1598:with compact support, i.e.
1381:
1307:if there is a neighborhood
342:that is locally given by a
10:
3360:
2432:Lagrange inversion theorem
2282:the following bound holds
1927:the following bound holds
1854:{\displaystyle K\subset D}
790:in which the coefficients
585:{\displaystyle x_{0}\in D}
355:complex analytic functions
29:
18:
3111:. Boca Raton: CRC Press.
2437:Any analytic function is
1372:complex analytic function
836:are real numbers and the
359:infinitely differentiable
228:Geometric function theory
174:Cauchy's integral formula
164:Cauchy's integral theorem
3074:in some neighborhood of
3051:of the complex variable
2963:
2933:CauchyâRiemann equations
2907:. This can be proved by
2610:) = 0 for all
2415:The sums, products, and
2219:there exists a constant
2074:be an open set, and let
1861:there exists a constant
1832:is smooth and for every
1658:, cannot be analytic on
1463:hypergeometric functions
136:CauchyâRiemann equations
19:Not to be confused with
3217:Krantz & Parks 2002
3140:Krantz & Parks 2002
3031:. McGraw-Hill. p.
2948:Quasi-analytic function
2634:. This is known as the
1440:trigonometric functions
351:real analytic functions
121:Complex-valued function
3198:encyclopediaofmath.org
3171:10.3792/pja/1195524081
3001:
2785:
2681:
2555:
2511:
2473:
2400:
2276:
2233:
2213:
2193:and for every compact
2187:
2145:
2125:
2102:
2068:
2016:
1921:
1901:
1900:{\displaystyle x\in K}
1875:
1855:
1826:
1804:
1784:
1756:
1733:
1713:
1681:
1652:
1592:
1565:
1536:
1361:
1341:
1321:
1301:
1281:
1258:
1226:
1185:
1162:
1135:
1115:
1083:
1003:
952:
914:
887:
867:
830:
781:
637:
586:
549:
522:
486:
451:
420:
389:
292:Mathematics portal
3002:
3000:{\displaystyle x_{0}}
2915:Domains of holomorphy
2881:radius of convergence
2786:
2682:
2556:
2512:
2474:
2401:
2277:
2234:
2214:
2188:
2146:
2126:
2103:
2069:
2027:holomorphic functions
2017:
1922:
1902:
1876:
1856:
1827:
1805:
1785:
1757:
1734:
1714:
1682:
1653:
1593:
1566:
1537:
1362:
1342:
1322:
1302:
1282:
1259:
1227:
1186:
1163:
1161:{\displaystyle x_{0}}
1141:in a neighborhood of
1136:
1116:
1084:
983:
953:
951:{\displaystyle x_{0}}
915:
913:{\displaystyle x_{0}}
893:in a neighborhood of
888:
868:
831:
782:
617:
587:
550:
523:
508:Formally, a function
487:
485:{\displaystyle x_{0}}
452:
450:{\displaystyle x_{0}}
421:
419:{\displaystyle x_{0}}
390:
388:{\displaystyle x_{0}}
244:Augustin-Louis Cauchy
46:Mathematical analysis
36:Window function (SQL)
2984:
2943:PaleyâWiener theorem
2938:Holomorphic function
2735:
2709:holomorphic function
2660:
2529:
2487:
2455:
2289:
2243:
2223:
2197:
2155:
2135:
2131:is real analytic on
2115:
2078:
2043:
1931:
1911:
1885:
1881:such that for every
1865:
1839:
1816:
1794:
1766:
1746:
1723:
1703:
1662:
1602:
1582:
1546:
1517:
1418:exponential function
1392:elementary functions
1370:The definition of a
1351:
1331:
1311:
1291:
1271:
1236:
1195:
1191:is often denoted by
1175:
1145:
1125:
1114:{\displaystyle f(x)}
1096:
965:
935:
897:
877:
866:{\displaystyle f(x)}
848:
794:
599:
563:
539:
512:
494:Weierstrass function
469:
434:
403:
372:
254:Carl Friedrich Gauss
189:Isolated singularity
131:Holomorphic function
32:holomorphic function
3315:"Analytic Function"
3293:"Analytic function"
2978:uniform convergence
2726:Liouville's theorem
2706:is synonymous with
2698:. Consequently, in
2588:connected component
2271:
1629:
349:. There exist both
141:Formal power series
103:Unit complex number
21:analytic expression
3344:Analytic functions
3312:Weisstein, Eric W.
3070:but at each point
2997:
2781:
2677:
2646:degrees of freedom
2614:and this sequence
2580:accumulation point
2551:
2507:
2469:
2396:
2307:
2272:
2252:
2229:
2209:
2183:
2141:
2121:
2098:
2064:
2012:
1917:
1897:
1871:
1851:
1822:
1800:
1780:
1752:
1729:
1709:
1677:
1648:
1611:
1588:
1561:
1532:
1435:(real or complex).
1367:is real analytic.
1357:
1337:
1317:
1297:
1277:
1254:
1222:
1181:
1158:
1131:
1111:
1079:
948:
910:
883:
863:
826:
777:
582:
545:
518:
482:
447:
416:
385:
219:Laplace's equation
199:Argument principle
3251:978-0-387-90328-6
2851:| â„ 1.
2776:
2704:analytic function
2622:in the domain of
2345:
2292:
2232:{\displaystyle C}
2144:{\displaystyle U}
2124:{\displaystyle f}
1971:
1920:{\displaystyle k}
1874:{\displaystyle C}
1825:{\displaystyle f}
1803:{\displaystyle D}
1755:{\displaystyle f}
1732:{\displaystyle D}
1712:{\displaystyle f}
1591:{\displaystyle f}
1503:complex conjugate
1496:Piecewise defined
1457:special functions
1360:{\displaystyle f}
1340:{\displaystyle x}
1320:{\displaystyle D}
1300:{\displaystyle x}
1280:{\displaystyle f}
1184:{\displaystyle D}
1134:{\displaystyle x}
1048:
886:{\displaystyle x}
548:{\displaystyle D}
521:{\displaystyle f}
336:analytic function
328:
327:
214:Harmonic function
126:Analytic function
112:Complex functions
98:Complex conjugate
3351:
3325:
3324:
3306:
3281:
3264:Parks, Harold R.
3255:
3220:
3214:
3208:
3207:
3205:
3204:
3190:
3184:
3183:
3173:
3149:
3143:
3137:
3131:
3130:
3102:
3096:
3095:
3030:
3020:
3008:
3006:
3004:
3003:
2998:
2996:
2995:
2974:
2875:) diverges for |
2846:
2790:
2788:
2787:
2782:
2777:
2775:
2768:
2767:
2754:
2700:complex analysis
2686:
2684:
2683:
2678:
2676:
2675:
2670:
2669:
2636:identity theorem
2560:
2558:
2557:
2552:
2541:
2540:
2523:Morera's theorem
2516:
2514:
2513:
2508:
2506:
2478:
2476:
2475:
2470:
2468:
2405:
2403:
2402:
2397:
2389:
2388:
2381:
2373:
2360:
2356:
2346:
2344:
2343:
2342:
2329:
2325:
2324:
2314:
2306:
2281:
2279:
2278:
2273:
2270:
2265:
2257:
2238:
2236:
2235:
2230:
2218:
2216:
2215:
2210:
2192:
2190:
2189:
2184:
2173:
2172:
2150:
2148:
2147:
2142:
2130:
2128:
2127:
2122:
2107:
2105:
2104:
2099:
2097:
2073:
2071:
2070:
2065:
2063:
2062:
2057:
2021:
2019:
2018:
2013:
2005:
2004:
1986:
1982:
1972:
1970:
1969:
1968:
1955:
1951:
1950:
1940:
1926:
1924:
1923:
1918:
1906:
1904:
1903:
1898:
1880:
1878:
1877:
1872:
1860:
1858:
1857:
1852:
1831:
1829:
1828:
1823:
1809:
1807:
1806:
1801:
1789:
1787:
1786:
1781:
1779:
1761:
1759:
1758:
1753:
1738:
1736:
1735:
1730:
1718:
1716:
1715:
1710:
1686:
1684:
1683:
1678:
1676:
1675:
1670:
1657:
1655:
1654:
1649:
1644:
1643:
1638:
1628:
1623:
1618:
1617:
1597:
1595:
1594:
1589:
1570:
1568:
1567:
1562:
1560:
1559:
1554:
1541:
1539:
1538:
1533:
1531:
1530:
1525:
1468:Bessel functions
1424:close enough to
1411:Maclaurin series
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1326:
1324:
1323:
1318:
1306:
1304:
1303:
1298:
1286:
1284:
1283:
1278:
1263:
1261:
1260:
1255:
1253:
1252:
1246:
1245:
1231:
1229:
1228:
1223:
1212:
1211:
1205:
1204:
1190:
1188:
1187:
1182:
1167:
1165:
1164:
1159:
1157:
1156:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1088:
1086:
1085:
1080:
1078:
1077:
1068:
1067:
1049:
1047:
1039:
1035:
1034:
1022:
1021:
1005:
1002:
997:
957:
955:
954:
949:
947:
946:
919:
917:
916:
911:
909:
908:
892:
890:
889:
884:
872:
870:
869:
864:
835:
833:
832:
827:
819:
818:
806:
805:
786:
784:
783:
778:
770:
769:
760:
759:
741:
740:
725:
724:
706:
705:
693:
692:
680:
679:
674:
670:
669:
668:
647:
646:
636:
631:
591:
589:
588:
583:
575:
574:
554:
552:
551:
546:
527:
525:
524:
519:
491:
489:
488:
483:
481:
480:
456:
454:
453:
448:
446:
445:
425:
423:
422:
417:
415:
414:
394:
392:
391:
386:
384:
383:
320:
313:
306:
290:
289:
274:Karl Weierstrass
269:Bernhard Riemann
259:Jacques Hadamard
88:Imaginary number
68:
58:Complex analysis
52:
50:Complex analysis
41:
40:
3359:
3358:
3354:
3353:
3352:
3350:
3349:
3348:
3334:
3333:
3291:
3288:
3278:
3252:
3234:Conway, John B.
3229:
3224:
3223:
3215:
3211:
3202:
3200:
3192:
3191:
3187:
3150:
3146:
3138:
3134:
3119:
3103:
3099:
3080:
3065:
3043:
3021:
3017:
3012:
3011:
2991:
2987:
2985:
2982:
2981:
2975:
2971:
2966:
2929:
2920:pseudoconvexity
2893:
2832:
2830:
2811:
2804:
2798:around a point
2763:
2759:
2758:
2753:
2736:
2733:
2732:
2718:
2671:
2665:
2664:
2663:
2661:
2658:
2657:
2654:
2609:
2595:
2536:
2532:
2530:
2527:
2526:
2502:
2488:
2485:
2484:
2464:
2456:
2453:
2452:
2412:
2377:
2369:
2368:
2364:
2338:
2334:
2330:
2320:
2316:
2315:
2313:
2312:
2308:
2296:
2290:
2287:
2286:
2266:
2258:
2253:
2244:
2241:
2240:
2224:
2221:
2220:
2198:
2195:
2194:
2168:
2164:
2156:
2153:
2152:
2151:if and only if
2136:
2133:
2132:
2116:
2113:
2112:
2093:
2079:
2076:
2075:
2058:
2053:
2052:
2044:
2041:
2040:
1994:
1990:
1964:
1960:
1956:
1946:
1942:
1941:
1939:
1938:
1934:
1932:
1929:
1928:
1912:
1909:
1908:
1886:
1883:
1882:
1866:
1863:
1862:
1840:
1837:
1836:
1817:
1814:
1813:
1795:
1792:
1791:
1790:which contains
1775:
1767:
1764:
1763:
1762:to an open set
1747:
1744:
1743:
1724:
1721:
1720:
1704:
1701:
1700:
1694:
1671:
1666:
1665:
1663:
1660:
1659:
1639:
1634:
1633:
1624:
1619:
1613:
1612:
1603:
1600:
1599:
1583:
1580:
1579:
1555:
1550:
1549:
1547:
1544:
1543:
1526:
1521:
1520:
1518:
1515:
1514:
1490:complex numbers
1473:gamma functions
1448:power functions
1430:
1384:
1352:
1349:
1348:
1332:
1329:
1328:
1312:
1309:
1308:
1292:
1289:
1288:
1272:
1269:
1268:
1247:
1241:
1240:
1239:
1237:
1234:
1233:
1206:
1200:
1199:
1198:
1196:
1193:
1192:
1176:
1173:
1172:
1152:
1148:
1146:
1143:
1142:
1126:
1123:
1122:
1097:
1094:
1093:
1073:
1069:
1063:
1059:
1040:
1030:
1026:
1011:
1007:
1006:
1004:
998:
987:
966:
963:
962:
942:
938:
936:
933:
932:
904:
900:
898:
895:
894:
878:
875:
874:
849:
846:
845:
814:
810:
801:
797:
795:
792:
791:
765:
761:
755:
751:
736:
732:
720:
716:
701:
697:
688:
684:
675:
664:
660:
653:
649:
648:
642:
638:
632:
621:
600:
597:
596:
592:one can write
570:
566:
564:
561:
560:
540:
537:
536:
513:
510:
509:
506:
498:Fabius function
476:
472:
470:
467:
466:
441:
437:
435:
432:
431:
410:
406:
404:
401:
400:
379:
375:
373:
370:
369:
324:
284:
194:Residue theorem
169:Local primitive
159:Zeros and poles
74:Complex numbers
44:
39:
28:
25:analytic signal
17:
12:
11:
5:
3357:
3347:
3346:
3332:
3331:
3326:
3307:
3287:
3286:External links
3284:
3283:
3282:
3276:
3260:Krantz, Steven
3256:
3250:
3228:
3225:
3222:
3221:
3209:
3185:
3144:
3132:
3117:
3097:
3078:
3063:
3041:
3014:
3013:
3010:
3009:
2994:
2990:
2968:
2967:
2965:
2962:
2961:
2960:
2955:
2950:
2945:
2940:
2935:
2928:
2925:
2924:
2923:
2912:
2892:
2889:
2867: = ±
2847:diverges for |
2828:
2809:
2802:
2792:
2791:
2780:
2774:
2771:
2766:
2762:
2757:
2752:
2749:
2746:
2743:
2740:
2717:
2714:
2674:
2668:
2653:
2650:
2605:
2593:
2575:
2574:
2550:
2547:
2544:
2539:
2535:
2505:
2501:
2498:
2495:
2492:
2467:
2463:
2460:
2446:
2435:
2420:
2411:
2408:
2407:
2406:
2395:
2392:
2387:
2384:
2380:
2376:
2372:
2367:
2363:
2359:
2355:
2352:
2349:
2341:
2337:
2333:
2328:
2323:
2319:
2311:
2305:
2302:
2299:
2295:
2269:
2264:
2261:
2256:
2251:
2248:
2228:
2208:
2205:
2202:
2182:
2179:
2176:
2171:
2167:
2163:
2160:
2140:
2120:
2096:
2092:
2089:
2086:
2083:
2061:
2056:
2051:
2048:
2023:
2022:
2011:
2008:
2003:
2000:
1997:
1993:
1989:
1985:
1981:
1978:
1975:
1967:
1963:
1959:
1954:
1949:
1945:
1937:
1916:
1896:
1893:
1890:
1870:
1850:
1847:
1844:
1821:
1811:
1799:
1778:
1774:
1771:
1751:
1740:
1728:
1708:
1693:
1690:
1689:
1688:
1674:
1669:
1647:
1642:
1637:
1632:
1627:
1622:
1616:
1610:
1607:
1587:
1572:
1558:
1553:
1529:
1524:
1509: →
1499:
1493:
1486:absolute value
1478:
1477:
1476:
1475:
1470:
1465:
1453:
1452:
1451:
1436:
1428:
1414:
1390:The following
1383:
1380:
1356:
1336:
1316:
1296:
1276:
1251:
1244:
1221:
1218:
1215:
1210:
1203:
1180:
1155:
1151:
1130:
1110:
1107:
1104:
1101:
1090:
1089:
1076:
1072:
1066:
1062:
1058:
1055:
1052:
1046:
1043:
1038:
1033:
1029:
1025:
1020:
1017:
1014:
1010:
1001:
996:
993:
990:
986:
982:
979:
976:
973:
970:
958:in its domain
945:
941:
927:such that the
907:
903:
882:
862:
859:
856:
853:
825:
822:
817:
813:
809:
804:
800:
788:
787:
776:
773:
768:
764:
758:
754:
750:
747:
744:
739:
735:
731:
728:
723:
719:
715:
712:
709:
704:
700:
696:
691:
687:
683:
678:
673:
667:
663:
659:
656:
652:
645:
641:
635:
630:
627:
624:
620:
616:
613:
610:
607:
604:
581:
578:
573:
569:
544:
517:
505:
502:
479:
475:
444:
440:
413:
409:
382:
378:
326:
325:
323:
322:
315:
308:
300:
297:
296:
295:
294:
279:
278:
277:
276:
271:
266:
261:
256:
251:
249:Leonhard Euler
246:
238:
237:
231:
230:
224:
223:
222:
221:
216:
211:
206:
201:
196:
191:
186:
184:Laurent series
181:
179:Winding number
176:
171:
166:
161:
153:
152:
146:
145:
144:
143:
138:
133:
128:
123:
115:
114:
108:
107:
106:
105:
100:
95:
90:
85:
77:
76:
70:
69:
61:
60:
54:
53:
15:
9:
6:
4:
3:
2:
3356:
3345:
3342:
3341:
3339:
3330:
3327:
3322:
3321:
3316:
3313:
3308:
3304:
3300:
3299:
3294:
3290:
3289:
3279:
3277:0-8176-4264-1
3273:
3269:
3265:
3261:
3257:
3253:
3247:
3243:
3239:
3235:
3231:
3230:
3218:
3213:
3199:
3195:
3189:
3181:
3177:
3172:
3167:
3163:
3159:
3155:
3148:
3142:, p. 15.
3141:
3136:
3128:
3124:
3120:
3118:0-8493-8273-4
3114:
3110:
3109:
3101:
3094:
3092:
3088:
3084:
3077:
3073:
3069:
3062:
3058:
3054:
3050:
3044:
3042:0-07-010855-2
3038:
3034:
3029:
3028:
3019:
3015:
2992:
2988:
2979:
2976:This implies
2973:
2969:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2921:
2916:
2913:
2910:
2906:
2902:
2901:
2900:
2898:
2888:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2857:
2852:
2850:
2844:
2840:
2836:
2827:
2823:
2819:
2815:
2808:
2801:
2797:
2778:
2772:
2769:
2764:
2760:
2755:
2750:
2744:
2738:
2731:
2730:
2729:
2727:
2724:According to
2722:
2713:
2711:
2710:
2705:
2701:
2697:
2692:
2690:
2649:
2647:
2642:
2639:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2608:
2604:
2600:
2596:
2589:
2585:
2581:
2572:
2568:
2567:supremum norm
2564:
2533:
2524:
2520:
2519:Fréchet space
2493:
2490:
2482:
2461:
2451:
2447:
2444:
2440:
2436:
2433:
2429:
2425:
2421:
2418:
2414:
2413:
2393:
2390:
2385:
2382:
2374:
2365:
2361:
2357:
2350:
2339:
2335:
2326:
2321:
2309:
2303:
2300:
2297:
2285:
2284:
2283:
2267:
2262:
2259:
2249:
2246:
2226:
2206:
2203:
2200:
2177:
2165:
2161:
2158:
2138:
2118:
2109:
2087:
2084:
2081:
2059:
2049:
2046:
2037:
2035:
2030:
2028:
2009:
2006:
2001:
1998:
1995:
1991:
1987:
1983:
1976:
1965:
1961:
1957:
1952:
1947:
1943:
1935:
1914:
1894:
1891:
1888:
1868:
1848:
1845:
1842:
1835:
1819:
1812:
1797:
1772:
1769:
1749:
1741:
1726:
1706:
1699:
1698:
1697:
1672:
1640:
1620:
1608:
1605:
1585:
1577:
1573:
1556:
1527:
1512:
1508:
1504:
1500:
1497:
1494:
1491:
1487:
1483:
1482:
1481:
1474:
1471:
1469:
1466:
1464:
1461:
1460:
1458:
1454:
1449:
1445:
1441:
1437:
1434:
1427:
1423:
1419:
1415:
1412:
1408:
1404:
1400:
1396:
1395:
1393:
1389:
1388:
1387:
1379:
1377:
1373:
1368:
1354:
1334:
1314:
1294:
1274:
1265:
1249:
1232:, or just by
1216:
1208:
1178:
1170:
1153:
1149:
1128:
1105:
1099:
1092:converges to
1074:
1064:
1060:
1056:
1053:
1044:
1041:
1031:
1027:
1015:
1008:
994:
991:
988:
984:
980:
974:
968:
961:
960:
959:
943:
939:
931:at any point
930:
929:Taylor series
926:
921:
905:
901:
880:
857:
851:
843:
839:
823:
820:
815:
811:
807:
802:
798:
774:
771:
766:
756:
752:
748:
745:
737:
733:
729:
721:
717:
713:
710:
702:
698:
694:
689:
685:
681:
676:
671:
665:
661:
657:
654:
650:
643:
639:
628:
625:
622:
618:
614:
608:
602:
595:
594:
593:
579:
576:
571:
567:
558:
542:
535:
531:
530:real analytic
515:
501:
499:
495:
477:
473:
464:
463:Taylor series
460:
459:Taylor series
442:
438:
429:
411:
407:
398:
380:
376:
367:
366:Taylor series
362:
360:
356:
352:
348:
345:
341:
337:
333:
321:
316:
314:
309:
307:
302:
301:
299:
298:
293:
288:
283:
282:
281:
280:
275:
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
241:
240:
239:
236:
233:
232:
229:
226:
225:
220:
217:
215:
212:
210:
209:Schwarz lemma
207:
205:
204:Conformal map
202:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
156:
155:
154:
151:
148:
147:
142:
139:
137:
134:
132:
129:
127:
124:
122:
119:
118:
117:
116:
113:
110:
109:
104:
101:
99:
96:
94:
93:Complex plane
91:
89:
86:
84:
81:
80:
79:
78:
75:
72:
71:
67:
63:
62:
59:
56:
55:
51:
47:
43:
42:
37:
33:
26:
22:
3318:
3296:
3267:
3237:
3212:
3201:. Retrieved
3197:
3188:
3164:(3): 90â93.
3161:
3157:
3147:
3135:
3107:
3100:
3090:
3089:. The term
3086:
3082:
3075:
3071:
3067:
3060:
3056:
3052:
3048:
3046:
3026:
3018:
2972:
2897:power series
2894:
2876:
2872:
2868:
2864:
2860:
2853:
2848:
2842:
2838:
2834:
2825:
2806:
2799:
2793:
2723:
2719:
2707:
2703:
2693:
2655:
2643:
2640:
2631:
2627:
2623:
2619:
2611:
2606:
2602:
2591:
2576:
2571:Banach space
2480:
2442:
2417:compositions
2110:
2038:
2031:
2024:
1695:
1510:
1506:
1479:
1432:
1425:
1421:
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1371:
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397:neighborhood
363:
354:
350:
347:power series
335:
329:
150:Basic theory
125:
49:
3091:holomorphic
3047:A function
2702:, the term
2630:containing
2618:to a point
2582:inside its
1834:compact set
1399:polynomials
1376:holomorphic
1267:A function
559:if for any
504:Definitions
332:mathematics
264:Kiyoshi Oka
83:Real number
3227:References
3203:2020-08-30
2525:. The set
2479:, the set
2428:derivative
2424:reciprocal
1446:, and the
842:convergent
399:for every
344:convergent
3320:MathWorld
3303:EMS Press
3180:0021-4280
3059:at point
2673:∞
2616:converges
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2500:→
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2391:α
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2322:α
2318:∂
2301:∈
2260:≥
2250:∈
2247:α
2204:⊆
2170:∞
2162:∈
2091:→
2050:⊂
1988:≤
1892:∈
1846:⊂
1773:⊂
1626:∞
1609:∈
1505:function
1444:logarithm
1347:on which
1250:ω
1209:ω
1169:pointwise
1057:−
1000:∞
985:∑
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775:⋯
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714:−
658:−
634:∞
619:∑
577:∈
557:real line
3338:Category
3266:(2002).
3236:(1978).
3127:28890674
3057:analytic
2927:See also
2905:discrete
2856:open set
2818:interval
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2448:For any
1382:Examples
534:open set
340:function
3305:, 2001
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555:in the
496:or the
426:in its
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2439:smooth
1574:Other
838:series
532:on an
428:domain
368:about
235:People
2964:Notes
2569:is a
2517:is a
2111:Then
1455:Most
338:is a
334:, an
3272:ISBN
3246:ISBN
3176:ISSN
3123:OCLC
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