276:
55:
1601:
2305:
is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in
1492:
2021:
1120:
1796:
1585:
is a function that is meromorphic on the whole
Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.
1966:
1252:
1343:
1659:
1592:
is meromorphic on the whole
Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.
2221:
714:
1416:
1163:
306:
2293:
2253:
2098:
1012:
906:
2185:
2052:
1890:
828:
778:
744:
645:
1858:
1829:
1692:
1543:
1373:
858:
2344:
299:
958:(see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The
2418:
2399:
292:
157:
1436:
1975:
1032:
502:
1708:
2428:
124:
162:
152:
2354:
544:
387:
481:, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole
2364:
2334:
490:
2479:
2129:
1906:
328:
216:
2307:
2339:
1175:
1272:
520:
332:
2374:
383:
2369:
1622:
959:
344:
340:
232:
34:
2190:
2159:
to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point
661:
275:
207:
2436:
548:
509:
407:
379:
242:
177:
119:
2349:
1386:
1132:
951:. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.
129:
91:
2359:
2113:
2319:
2109:
1423:
970:
918:, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.
575:
361:
280:
187:
2266:
2226:
2452:
2414:
2395:
1589:
1501:
532:
486:
202:
114:
86:
2141:
2068:
987:
875:
320:
262:
257:
247:
223:
138:
109:
100:
76:
46:
2170:
2030:
1972:
is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at
1863:
806:
756:
722:
623:
2137:
2132:
of dimension one (over the complex numbers). The simplest examples of such curves are the
1834:
1805:
1668:
1605:
954:
A meromorphic function may have infinitely many zeros and poles. This is the case for the
182:
1518:
1348:
833:
2387:
2324:
1665:
is meromorphic on the whole
Riemann sphere. It has a pole of order 1 or simple pole at
1575:
1019:
955:
915:
336:
237:
172:
167:
62:
2473:
2329:
2298:
2133:
2024:
1571:
648:
536:
482:
197:
192:
81:
962:
is also meromorphic in the whole complex plane, with a single pole of order 1 at
2145:
1600:
252:
71:
17:
1556:
969:. Its zeros in the left halfplane are all the negative even integers, and the
929:
and the symmetry between them, it is often useful to consider a pole of order
1422:
if it is meromorphic in some neighbourhood of infinity (that is outside some
2460:
2140:. This extension is done by transferring structures and properties through
2455:
2104:
has a single pole at infinity of order 1, and a single zero at the origin.
914:
This characterization of zeros and poles implies that zeros and poles are
54:
513:
411:
1802:
is meromorphic on the whole
Riemann sphere. It has a pole of order 2 at
2112:. For a general discussion of zeros and poles of such functions, see
1604:
A polynomial of degree 9 has a pole of order 9 at â, here plotted by
2124:
The concept of zeros and poles extends naturally to functions on a
1599:
620:
is a function that is meromorphic in a neighbourhood of a point
493:
of its poles equals the sum of the multiplicities of its zeros.
925:
of zeros and poles being defined as a non-negative number
2223:
is holomorphic (resp. meromorphic) in a neighbourhood of
1487:{\displaystyle \lim _{z\to \infty }{\frac {f(z)}{z^{n}}}}
2016:{\displaystyle z=2\pi ni{\text{ for }}n\in \mathbb {Z} }
1115:{\displaystyle f(z)=\sum _{k\geq -n}a_{k}(z-z_{0})^{k},}
1345:, there is no principal part), one has a zero of order
1791:{\displaystyle f(z)={\frac {z+2}{(z-5)^{2}(z+7)^{3}}}}
2269:
2229:
2193:
2173:
2071:
2033:
1978:
1909:
1866:
1837:
1808:
1711:
1671:
1625:
1521:
1439:
1389:
1351:
1275:
1178:
1135:
1035:
990:
878:
836:
809:
759:
746:(this is a consequence of the analytic property). If
725:
664:
626:
2287:
2247:
2215:
2179:
2092:
2046:
2015:
1960:
1884:
1852:
1823:
1790:
1686:
1653:
1537:
1486:
1410:
1367:
1337:
1246:
1157:
1114:
1006:
900:
852:
822:
772:
738:
708:
639:
973:is the conjecture that all other zeros are along
719:is holomorphic and nonzero in a neighbourhood of
1441:
1574:extended by a point at infinity is called the
559:has a neighbourhood such that at least one of
2114:Poleâzero plot § Continuous-time systems
872:are terms used for zeroes and poles of order
300:
8:
2433:Applied and Computational Complex Analysis 1
2108:All above examples except for the third are
1961:{\displaystyle f(z)={\frac {z-4}{e^{z}-1}}}
531:. Equivalently, it is holomorphic if it is
307:
293:
29:
2268:
2228:
2204:
2192:
2172:
2070:
2038:
2032:
2009:
2008:
1997:
1977:
1943:
1925:
1908:
1865:
1836:
1807:
1779:
1757:
1727:
1710:
1670:
1641:
1624:
1530:
1522:
1520:
1476:
1456:
1444:
1438:
1388:
1360:
1352:
1350:
1328:
1320:
1319:
1309:
1289:
1281:
1280:
1274:
1237:
1229:
1225:
1215:
1195:
1187:
1183:
1177:
1140:
1134:
1103:
1093:
1074:
1055:
1034:
995:
989:
911:is sometimes used synonymously to order.
887:
879:
877:
845:
837:
835:
814:
808:
764:
758:
730:
724:
688:
678:
663:
631:
625:
339:variable. It is the simplest type of non-
1497:exists and is a nonzero complex number.
1247:{\displaystyle a_{-|n|}(z-z_{0})^{-|n|}}
1026:(the terms with negative index values):
543:, and converges to the function in some
222:
215:
137:
99:
61:
45:
1338:{\displaystyle a_{|n|}(z-z_{0})^{|n|}}
7:
2411:Functions of One Complex Variable II
2345:Hurwitz's theorem (complex analysis)
2392:Functions of One Complex Variable I
2155:be a function from a complex curve
1654:{\displaystyle f(z)={\frac {3}{z}}}
2023:. This can be seen by writing the
1451:
25:
2216:{\displaystyle f\circ \phi ^{-1}}
1892:and a quadruple zero at infinity.
709:{\displaystyle (z-z_{0})^{n}f(z)}
473:. This induces a duality between
1258:terms), one has a pole of order
274:
53:
2325:Control theory § Stability
1014:a nonzero meromorphic function
651:, then there exists an integer
2279:
2273:
2239:
2233:
2081:
2075:
1919:
1913:
1776:
1763:
1754:
1741:
1721:
1715:
1694:and a simple zero at infinity.
1635:
1629:
1531:
1523:
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1462:
1448:
1405:
1399:
1393:
1361:
1353:
1329:
1321:
1316:
1296:
1290:
1282:
1238:
1230:
1222:
1202:
1196:
1188:
1100:
1080:
1045:
1039:
984:In a neighbourhood of a point
888:
880:
846:
838:
703:
697:
685:
665:
503:function of a complex variable
1:
2259:is a pole or a zero of order
1411:{\displaystyle z\mapsto f(z)}
1158:{\displaystyle a_{-n}\neq 0.}
323:(a branch of mathematics), a
547:of the point. A function is
2355:Nyquist stability criterion
1426:), and there is an integer
425:there is a neighborhood of
27:Concept in complex analysis
2496:
2365:Residue (complex analysis)
2335:Filter (signal processing)
1860:. It has a simple zero at
579:of a meromorphic function
2288:{\displaystyle \phi (z).}
2248:{\displaystyle \phi (z).}
2130:complex analytic manifold
1831:and a pole of order 3 at
1254:, the principal part has
539:exists at every point of
429:in which at least one of
217:Geometric function theory
163:Cauchy's integral formula
153:Cauchy's integral theorem
2409:Conway, John B. (1995).
2263:if the same is true for
356:is a pole of a function
347:). Technically, a point
343:of such a function (see
125:CauchyâRiemann equations
1420:meromorphic at infinity
333:complex-valued function
110:Complex-valued function
2289:
2249:
2217:
2181:
2094:
2093:{\displaystyle f(z)=z}
2048:
2017:
1962:
1886:
1854:
1825:
1792:
1688:
1655:
1609:
1608:of the Riemann sphere.
1539:
1515:, and a zero of order
1488:
1412:
1369:
1339:
1248:
1159:
1116:
1008:
1007:{\displaystyle z_{0},}
902:
901:{\displaystyle |n|=1.}
854:
824:
774:
740:
710:
641:
570:is holomorphic in it.
489:, then the sum of the
384:complex differentiable
281:Mathematics portal
2437:John Wiley & Sons
2290:
2250:
2218:
2182:
2180:{\displaystyle \phi }
2144:, which are analytic
2095:
2049:
2047:{\displaystyle e^{z}}
2018:
1963:
1887:
1885:{\displaystyle z=-2,}
1855:
1826:
1793:
1689:
1656:
1603:
1563:has a pole of degree
1540:
1489:
1413:
1370:
1340:
1269:(the sum starts with
1249:
1172:(the sum starts with
1160:
1117:
1009:
960:Riemann zeta function
903:
855:
825:
823:{\displaystyle z_{0}}
775:
773:{\displaystyle z_{0}}
741:
739:{\displaystyle z_{0}}
711:
642:
640:{\displaystyle z_{0}}
345:essential singularity
341:removable singularity
327:is a certain type of
233:Augustin-Louis Cauchy
35:Mathematical analysis
2308:RiemannâRoch theorem
2267:
2227:
2191:
2171:
2167:if there is a chart
2151:More precisely, let
2069:
2031:
1976:
1907:
1864:
1853:{\displaystyle z=-7}
1835:
1824:{\displaystyle z=5,}
1806:
1709:
1687:{\displaystyle z=0,}
1669:
1623:
1519:
1437:
1387:
1349:
1273:
1176:
1133:
1033:
1022:with at most finite
988:
940:and a zero of order
876:
834:
807:
757:
723:
662:
624:
583:is a complex number
243:Carl Friedrich Gauss
178:Isolated singularity
120:Holomorphic function
2375:Sendov's conjecture
2340:GaussâLucas theorem
2301:, and the function
2120:Function on a curve
1538:{\displaystyle |n|}
1504:is a pole of order
1368:{\displaystyle |n|}
1129:is an integer, and
944:as a pole of order
933:as a zero of order
853:{\displaystyle |n|}
830:is a zero of order
417:if for every point
130:Formal power series
92:Unit complex number
2453:Weisstein, Eric W.
2320:Argument principle
2285:
2245:
2213:
2177:
2110:rational functions
2090:
2054:around the origin.
2044:
2013:
1958:
1882:
1850:
1821:
1788:
1684:
1651:
1610:
1535:
1500:In this case, the
1484:
1455:
1408:
1365:
1335:
1244:
1155:
1112:
1069:
1004:
971:Riemann hypothesis
898:
850:
820:
788:(or multiplicity)
770:
736:
706:
637:
555:if every point of
535:, that is, if its
527:at every point of
447:is meromorphic in
208:Laplace's equation
188:Argument principle
2000:
1956:
1786:
1649:
1590:rational function
1502:point at infinity
1482:
1440:
1051:
487:point at infinity
451:, then a zero of
317:
316:
203:Harmonic function
115:Analytic function
101:Complex functions
87:Complex conjugate
16:(Redirected from
2487:
2480:Complex analysis
2466:
2465:
2440:
2424:
2405:
2370:Rouché's theorem
2350:Marden's theorem
2304:
2297:If the curve is
2294:
2292:
2291:
2286:
2262:
2258:
2254:
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2222:
2220:
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2019:
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2012:
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1507:
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1471:
1457:
1454:
1429:
1417:
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1409:
1374:
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1201:
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1199:
1191:
1171:
1164:
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1161:
1156:
1148:
1147:
1128:
1121:
1119:
1118:
1113:
1108:
1107:
1098:
1097:
1079:
1078:
1068:
1018:is the sum of a
1017:
1013:
1011:
1010:
1005:
1000:
999:
980:
968:
950:
943:
939:
932:
928:
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734:
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569:
562:
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542:
530:
526:
523:with respect to
518:
507:
472:
465:
462:, and a pole of
461:
454:
450:
446:
440:is holomorphic.
439:
432:
428:
424:
420:
416:
405:
398:
377:
370:
364:of the function
359:
355:
321:complex analysis
309:
302:
295:
279:
278:
263:Karl Weierstrass
258:Bernhard Riemann
248:Jacques Hadamard
77:Imaginary number
57:
47:Complex analysis
41:
39:Complex analysis
30:
21:
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2494:
2490:
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2486:
2485:
2484:
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2451:
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2427:
2421:
2408:
2402:
2388:Conway, John B.
2386:
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2265:
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2260:
2256:
2225:
2224:
2200:
2189:
2188:
2169:
2168:
2164:
2160:
2156:
2152:
2138:Riemann surface
2122:
2067:
2066:
2034:
2029:
2028:
1999: for
1974:
1973:
1939:
1938:
1927:
1905:
1904:
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1804:
1803:
1775:
1753:
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1729:
1707:
1706:
1667:
1666:
1621:
1620:
1606:domain coloring
1598:
1582:
1564:
1560:
1555:For example, a
1546:
1517:
1516:
1509:
1505:
1472:
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991:
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926:
921:Because of the
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313:
273:
183:Residue theorem
158:Local primitive
148:Zeros and poles
63:Complex numbers
33:
28:
23:
22:
18:Poles and zeros
15:
12:
11:
5:
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2446:
2445:External links
2443:
2442:
2441:
2429:Henrici, Peter
2425:
2419:
2406:
2400:
2382:
2379:
2378:
2377:
2372:
2367:
2362:
2360:Poleâzero plot
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1576:Riemann sphere
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1154:
1151:
1146:
1143:
1139:
1123:
1122:
1111:
1106:
1102:
1096:
1092:
1088:
1085:
1082:
1077:
1073:
1067:
1064:
1061:
1058:
1054:
1050:
1047:
1044:
1041:
1038:
1024:principal part
1020:Laurent series
1003:
998:
994:
956:gamma function
897:
894:
890:
886:
882:
848:
844:
840:
817:
813:
767:
763:
733:
729:
717:
716:
705:
702:
699:
696:
691:
687:
681:
677:
673:
670:
667:
634:
630:
521:differentiable
498:
495:
491:multiplicities
395:
352:
315:
314:
312:
311:
304:
297:
289:
286:
285:
284:
283:
268:
267:
266:
265:
260:
255:
250:
245:
240:
238:Leonhard Euler
235:
227:
226:
220:
219:
213:
212:
211:
210:
205:
200:
195:
190:
185:
180:
175:
173:Laurent series
170:
168:Winding number
165:
160:
155:
150:
142:
141:
135:
134:
133:
132:
127:
122:
117:
112:
104:
103:
97:
96:
95:
94:
89:
84:
79:
74:
66:
65:
59:
58:
50:
49:
43:
42:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2492:
2481:
2478:
2477:
2475:
2463:
2462:
2457:
2454:
2449:
2448:
2444:
2438:
2434:
2430:
2426:
2422:
2420:0-387-94460-5
2416:
2412:
2407:
2403:
2401:0-387-90328-3
2397:
2393:
2389:
2385:
2384:
2380:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2330:Filter design
2328:
2326:
2323:
2321:
2318:
2317:
2313:
2311:
2309:
2300:
2295:
2282:
2276:
2270:
2242:
2236:
2230:
2208:
2205:
2201:
2197:
2194:
2174:
2149:
2147:
2143:
2139:
2135:
2134:complex plane
2131:
2127:
2126:complex curve
2119:
2117:
2115:
2111:
2103:
2087:
2084:
2078:
2072:
2065:
2064:
2063:
2062:
2058:
2057:
2039:
2035:
2026:
2025:Taylor series
2005:
2002:
1994:
1991:
1988:
1985:
1982:
1979:
1971:
1952:
1949:
1944:
1940:
1934:
1931:
1928:
1922:
1916:
1910:
1903:
1902:
1901:
1900:
1896:
1895:
1879:
1876:
1873:
1870:
1867:
1847:
1844:
1841:
1838:
1818:
1815:
1812:
1809:
1801:
1780:
1772:
1769:
1766:
1758:
1750:
1747:
1744:
1736:
1733:
1730:
1724:
1718:
1712:
1705:
1704:
1703:
1702:
1698:
1697:
1681:
1678:
1675:
1672:
1664:
1646:
1643:
1638:
1632:
1626:
1619:
1618:
1617:
1616:
1612:
1611:
1607:
1602:
1595:
1593:
1591:
1586:
1579:
1577:
1573:
1572:complex plane
1568:
1567:at infinity.
1558:
1553:
1549:
1527:
1512:
1503:
1498:
1477:
1473:
1465:
1459:
1445:
1433:
1432:
1431:
1425:
1421:
1402:
1396:
1390:
1378:
1376:
1357:
1325:
1310:
1306:
1302:
1299:
1286:
1277:
1266:
1234:
1226:
1216:
1212:
1208:
1205:
1192:
1184:
1180:
1169:
1152:
1149:
1144:
1141:
1137:
1109:
1104:
1094:
1090:
1086:
1083:
1075:
1071:
1065:
1062:
1059:
1056:
1052:
1048:
1042:
1036:
1029:
1028:
1027:
1025:
1021:
1001:
996:
992:
982:
978:
972:
966:
961:
957:
952:
949:
938:
924:
919:
917:
912:
910:
895:
892:
884:
871:
867:
842:
815:
811:
800:
787:
783:
765:
761:
750:
731:
727:
700:
694:
689:
679:
675:
671:
668:
658:
657:
656:
650:
649:complex plane
632:
628:
614:
611:
606:is a zero of
601:
595:
591:
578:
577:
571:
568:
550:
546:
545:neighbourhood
538:
537:Taylor series
534:
522:
515:
511:
504:
496:
494:
492:
488:
484:
483:complex plane
480:
476:
471:
466:is a zero of
460:
455:is a pole of
441:
438:
413:
409:
400:
394:
389:
388:neighbourhood
385:
381:
376:
369:
363:
351:
346:
342:
338:
334:
330:
326:
322:
310:
305:
303:
298:
296:
291:
290:
288:
287:
282:
277:
272:
271:
270:
269:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
230:
229:
228:
225:
221:
218:
214:
209:
206:
204:
201:
199:
198:Schwarz lemma
196:
194:
193:Conformal map
191:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
145:
144:
143:
140:
136:
131:
128:
126:
123:
121:
118:
116:
113:
111:
108:
107:
106:
105:
102:
98:
93:
90:
88:
85:
83:
82:Complex plane
80:
78:
75:
73:
70:
69:
68:
67:
64:
60:
56:
52:
51:
48:
44:
40:
36:
32:
31:
19:
2459:
2432:
2413:. Springer.
2410:
2394:. Springer.
2391:
2296:
2150:
2146:isomorphisms
2125:
2123:
2107:
2059:The function
1897:The function
1699:The function
1613:The function
1587:
1580:
1569:
1554:
1547:
1510:
1499:
1496:
1419:
1382:
1264:
1167:
1124:
1023:
983:
976:
964:
953:
947:
936:
922:
920:
913:
908:
869:
865:
798:
785:
781:
748:
718:
615:
609:
599:
593:
589:
574:
572:
566:
500:
478:
474:
469:
458:
442:
436:
401:
392:
374:
367:
349:
324:
318:
147:
139:Basic theory
38:
1430:such that
1383:A function
1379:At infinity
870:simple pole
866:Simple zero
655:such that
549:meromorphic
514:open domain
510:holomorphic
497:Definitions
408:meromorphic
402:A function
380:holomorphic
360:if it is a
329:singularity
253:Kiyoshi Oka
72:Real number
2381:References
2187:such that
2128:, that is
1559:of degree
1557:polynomial
1165:Again, if
587:such that
386:) in some
2461:MathWorld
2271:ϕ
2231:ϕ
2206:−
2202:ϕ
2198:∘
2175:ϕ
2006:∈
1989:π
1950:−
1932:−
1874:−
1845:−
1748:−
1452:∞
1449:→
1394:↦
1303:−
1262:, and if
1227:−
1209:−
1185:−
1150:≠
1142:−
1087:−
1063:−
1060:≥
1053:∑
672:−
519:if it is
485:plus the
2474:Category
2431:(1974).
2390:(1986).
2314:See also
2136:and the
1596:Examples
916:isolated
533:analytic
412:open set
2299:compact
979:) = 1/2
803:, then
753:, then
647:of the
337:complex
2456:"Pole"
2417:
2398:
2255:Then,
2142:charts
1588:Every
1550:< 0
1513:> 0
1170:> 0
1125:where
909:Degree
801:< 0
751:> 0
512:in an
410:in an
382:(i.e.
224:People
923:order
796:. If
786:order
780:is a
596:) = 0
479:poles
475:zeros
335:of a
331:of a
2415:ISBN
2396:ISBN
1570:The
1424:disk
868:and
782:pole
600:pole
598:. A
576:zero
563:and
477:and
433:and
371:and
362:zero
325:pole
2163:of
2027:of
1581:If
1545:if
1508:if
1442:lim
1418:is
1267:†0
975:Re(
967:= 1
860:of
792:of
784:of
616:If
602:of
551:in
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390:of
378:is
319:In
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896:1.
864:.
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608:1/
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565:1/
501:A
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399:.
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366:1/
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2082:)
2079:z
2076:(
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2040:z
2036:e
2010:Z
2003:n
1995:i
1992:n
1986:2
1983:=
1980:z
1953:1
1945:z
1941:e
1935:4
1929:z
1923:=
1920:)
1917:z
1914:(
1911:f
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1842:=
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1819:,
1816:5
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1773:7
1770:+
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1764:(
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1716:(
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1300:z
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1095:0
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997:0
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889:|
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862:f
847:|
843:n
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816:0
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799:n
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557:U
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