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2112:): the question, going back to the mathematics of the seventeenth century, of finding the curves that cross a given family of non-intersecting curves at right angles.
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were interchanged, the functions would not be harmonic conjugates, since the minus sign in the Cauchy–Riemann equations makes the relationship asymmetric.
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in order to fix the indeterminacy of the conjugate up to constants). This is well known in applications as (essentially) the
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1546:{\displaystyle {\partial u \over \partial y}=e^{x}\cos y,\quad {\partial ^{2}u \over \partial y^{2}}=-e^{x}\sin y,}
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1421:{\displaystyle {\partial u \over \partial x}=e^{x}\sin y,\quad {\partial ^{2}u \over \partial x^{2}}=e^{x}\sin y}
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and a transform relating their solutions), in this case linear; more complex transforms are of interest in
1162:. Conjugate harmonic functions (and the transform between them) are also one of the simplest examples of a
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is not zero) gives rise to a geometric property of harmonic conjugates. Clearly the harmonic conjugate of
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and their first-order partial derivatives satisfy the Cauchy-Riemann equations (2) throughout
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1822:{\displaystyle {\partial u \over \partial y}=-{\partial v \over \partial x}=e^{x}\cos y.}
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1734:{\displaystyle {\partial u \over \partial x}={\partial v \over \partial y}=e^{x}\sin y}
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37:"Conjugate function" redirects here. For the convex conjugate of a function, see
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admits a conjugated harmonic function if and only if the holomorphic function
1088:, and in any case it admits a conjugate locally at any point of its domain.
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So any harmonic function always admits a conjugate function whenever its
2108:(not the only solution, naturally, since we can take also functions of
1171:
653:
As an immediate consequence of the latter equivalent definition, if
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will also be orthogonal where they cross (away from the zeros of
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of the underlying holomorphic function; the contours on which
1945:{\displaystyle {\partial v \over \partial x}=-e^{x}\cos y}
1883:{\displaystyle {\partial v \over \partial y}=e^{x}\sin y}
384:
As a first consequence of the definition, they are both
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such that the Cauchy–Riemann equations are satisfied:
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of each other with respect to another pair of points
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704:{\displaystyle \Omega \subset \mathbb {R} ^{2},}
2176:(6th ed.). New York: McGraw-Hill. p.
768:for then the Cauchy–Riemann equations are just
121:{\displaystyle \Omega \subset \mathbb {R} ^{2}}
2168:Brown, James Ward; Churchill, Ruel V. (1996).
2126:There is an additional occurrence of the term
1622:) and is thus harmonic. Now suppose we have a
799:symmetry of the mixed second order derivatives
8:
2104:problem for the family of contours given by
951:{\displaystyle g(z):=u_{x}(x,y)-iu_{y}(x,y)}
1073:{\displaystyle \operatorname {Im} f(x+iy).}
2130:in mathematics, and more specifically in
2008:Observe that if the functions related to
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163:if and only if they are respectively the
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2051:are orthogonal. Conformality says that
1591:{\displaystyle \Delta u=\nabla ^{2}u=0}
128:is said to have a conjugate (function)
7:
354:{\displaystyle f(z):=u(x,y)+iv(x,y)}
30:For geometric conjugate points, see
1300:{\displaystyle u(x,y)=e^{x}\sin y.}
1247:For example, consider the function
240:{\displaystyle z:=x+iy\in \Omega .}
2172:Complex variables and applications
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1999:{\displaystyle v=-e^{x}\cos y+C.}
1154:; it is also a basic example in
1121:{\displaystyle \mathbb {R} ^{2}}
1099:on a simply connected region in
850:Therefore, a harmonic function
1480:
1361:
2271:Partial differential equations
2239:"Conjugate harmonic functions"
2116:Harmonic conjugate in geometry
2100:is a specific solution of the
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2122:Projective harmonic conjugate
1013:in which case a conjugate of
408:. Moreover, the conjugate of
32:Projective harmonic conjugate
2043:, and the lines of constant
673:is any harmonic function on
435:an additive constant. Also,
18:Conjugate harmonic functions
2244:Encyclopedia of Mathematics
1160:singular integral operators
1128:to its harmonic conjugate
1095:taking a harmonic function
2287:
2119:
790:{\displaystyle \Delta u=0}
36:
29:
2200:are harmonic in a domain
388:real-valued functions on
1952:which when solved gives
1006:{\displaystyle \Omega ,}
646:{\displaystyle \Omega .}
628:Cauchy–Riemann equations
431:if it exists, is unique
377:{\displaystyle \Omega .}
165:real and imaginary parts
2192:If two given functions
1611:{\displaystyle \Delta }
1192:orthogonal trajectories
579:{\displaystyle \Omega }
401:{\displaystyle \Omega }
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1946:
1884:
1823:
1735:
1651:
1650:{\displaystyle v(x,y)}
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1207:are constant cross at
1189:are related as having
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734:{\displaystyle -u_{y}}
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156:{\displaystyle v(x,y)}
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84:
83:{\displaystyle u(x,y)}
27:Concept in mathematics
2102:orthogonal trajectory
2031:(at points where the
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1158:, in connection with
1156:mathematical analysis
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761:{\displaystyle u_{x}}
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983:{\displaystyle f(z)}
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192:{\displaystyle f(z)}
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169:holomorphic function
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2144:harmonic conjugates
2132:projective geometry
2096:). That means that
2266:Harmonic functions
2214:harmonic conjugate
2128:harmonic conjugate
2029:analytic functions
1996:
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1229:potential function
1211:. In this regard,
1176:integrable systems
1164:Bäcklund transform
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511:{\displaystyle -u}
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424:{\displaystyle u,}
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361:is holomorphic on
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92:connected open set
80:
2025:conformal mapping
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1221:complex potential
1152:Hilbert transform
1143:) = 0 on a given
1026:{\displaystyle u}
863:{\displaystyle u}
666:{\displaystyle u}
619:{\displaystyle v}
599:{\displaystyle u}
559:{\displaystyle u}
539:{\displaystyle v}
488:{\displaystyle v}
468:{\displaystyle v}
448:{\displaystyle u}
280:{\displaystyle u}
260:{\displaystyle v}
16:(Redirected from
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2212:is said to be a
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1086:simply connected
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39:Convex conjugate
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2142:are said to be
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1237:stream function
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1033:is, of course,
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586:if and only if
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2233:Harmonic Ratio
2228:
2227:External links
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2120:Main article:
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1181:Geometrically
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2187:0-07-912147-0
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2158:) equals −1.
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2134:. Two points
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2059:
2054:
2050:
2047:and constant
2046:
2042:
2038:
2034:
2030:
2026:
2021:
2018:
2012:
2006:
1993:
1990:
1987:
1984:
1981:
1978:
1973:
1969:
1965:
1962:
1959:
1939:
1936:
1933:
1928:
1924:
1920:
1917:
1911:
1903:
1877:
1874:
1871:
1866:
1862:
1858:
1852:
1844:
1831:Simplifying,
1829:
1816:
1813:
1810:
1807:
1802:
1798:
1794:
1788:
1780:
1771:
1768:
1762:
1754:
1728:
1725:
1722:
1717:
1713:
1709:
1703:
1695:
1686:
1680:
1672:
1658:
1641:
1638:
1635:
1629:
1621:
1585:
1582:
1579:
1574:
1566:
1563:
1553:it satisfies
1540:
1537:
1534:
1531:
1526:
1522:
1518:
1515:
1507:
1503:
1494:
1489:
1477:
1474:
1471:
1468:
1463:
1459:
1455:
1449:
1441:
1415:
1412:
1409:
1404:
1400:
1396:
1388:
1384:
1375:
1370:
1358:
1355:
1352:
1349:
1344:
1340:
1336:
1330:
1322:
1307:
1294:
1291:
1288:
1285:
1280:
1276:
1272:
1266:
1263:
1260:
1254:
1242:
1240:
1238:
1234:
1230:
1226:
1222:
1219:would be the
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1193:
1188:
1184:
1179:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1146:
1139:
1135:
1131:
1113:
1098:
1094:
1089:
1087:
1083:
1067:
1061:
1058:
1055:
1052:
1046:
1043:
1040:
1020:
1000:
974:
968:
961:
942:
939:
936:
928:
924:
920:
917:
911:
908:
905:
897:
893:
889:
883:
877:
857:
837:
832:
829:
825:
821:
816:
813:
809:
800:
784:
781:
778:
753:
749:
726:
722:
718:
711:the function
698:
693:
683:
660:
640:
629:
613:
593:
553:
533:
521:
519:
505:
502:
482:
462:
442:
434:
418:
415:
387:
371:
345:
342:
339:
333:
330:
327:
321:
318:
315:
309:
306:
300:
294:
274:
254:
234:
228:
225:
222:
219:
216:
213:
210:
202:
183:
177:
170:
166:
147:
144:
141:
135:
113:
103:
93:
90:defined on a
74:
71:
68:
62:
55:
51:
47:
40:
33:
19:
2242:
2217:
2213:
2209:
2205:
2201:
2197:
2193:
2191:
2171:
2155:
2147:
2143:
2139:
2135:
2127:
2125:
2109:
2105:
2097:
2091:
2087:
2080:
2076:
2072:
2065:
2061:
2057:
2055:of constant
2048:
2044:
2040:
2036:
2027:property of
2022:
2016:
2010:
2007:
1830:
1659:
1308:
1246:
1232:
1224:
1216:
1212:
1209:right angles
1204:
1200:
1190:
1186:
1182:
1180:
1144:
1137:
1133:
1129:
1096:
1091:There is an
1090:
626:satisfy the
525:
43:
2152:cross ratio
522:Description
46:mathematics
2260:Categories
2162:References
2033:derivative
2249:EMS Press
2090: ′(
1982:
1966:−
1937:
1921:−
1909:∂
1901:∂
1875:
1850:∂
1842:∂
1811:
1786:∂
1778:∂
1772:−
1760:∂
1752:∂
1726:
1701:∂
1693:∂
1678:∂
1670:∂
1606:Δ
1571:∇
1561:Δ
1535:
1519:−
1500:∂
1486:∂
1472:
1447:∂
1439:∂
1413:
1381:∂
1367:∂
1353:
1328:∂
1320:∂
1289:
1044:
998:Ω
960:primitive
918:−
776:Δ
719:−
684:⊂
681:Ω
638:Ω
574:Ω
503:−
396:Ω
369:Ω
247:That is,
232:Ω
229:∈
203:variable
104:⊂
101:Ω
2053:contours
1243:Examples
1223:, where
1172:solitons
1093:operator
797:and the
386:harmonic
54:function
52:-valued
2251:, 2001
2150:if the
1618:is the
1235:is the
1227:is the
201:complex
199:of the
2184:
1309:Since
1082:domain
958:has a
1197:zeros
1166:(two
433:up to
167:of a
2196:and
2182:ISBN
2156:ABCD
2148:C, D
2138:and
2070:and
2023:The
2014:and
1890:and
1741:and
1428:and
1231:and
1203:and
1185:and
1174:and
1168:PDEs
606:and
50:real
48:, a
2216:of
2039:is
1979:cos
1934:cos
1872:sin
1808:cos
1723:sin
1532:sin
1469:cos
1410:sin
1350:sin
1286:sin
1084:is
990:in
630:in
566:in
287:if
44:In
2262::
2247:,
2241:,
2208:,
2190:.
2180:.
2178:61
2079:,
2064:,
1239:.
1217:iv
1215:+
1178:.
1041:Im
890::=
801:,
518:.
307::=
214::=
2220:.
2218:u
2210:v
2206:D
2202:D
2198:v
2194:u
2154:(
2140:B
2136:A
2110:v
2106:u
2098:v
2094:)
2092:z
2088:f
2083:)
2081:y
2077:x
2075:(
2073:v
2068:)
2066:y
2062:x
2060:(
2058:u
2049:y
2045:x
2041:y
2037:x
2017:v
2011:u
1994:.
1991:C
1988:+
1985:y
1974:x
1970:e
1963:=
1960:v
1940:y
1929:x
1925:e
1918:=
1912:x
1904:v
1878:y
1867:x
1863:e
1859:=
1853:y
1845:v
1817:.
1814:y
1803:x
1799:e
1795:=
1789:x
1781:v
1769:=
1763:y
1755:u
1729:y
1718:x
1714:e
1710:=
1704:y
1696:v
1687:=
1681:x
1673:u
1645:)
1642:y
1639:,
1636:x
1633:(
1630:v
1598:(
1586:0
1583:=
1580:u
1575:2
1567:=
1564:u
1541:,
1538:y
1527:x
1523:e
1516:=
1508:2
1504:y
1495:u
1490:2
1478:,
1475:y
1464:x
1460:e
1456:=
1450:y
1442:u
1416:y
1405:x
1401:e
1397:=
1389:2
1385:x
1376:u
1371:2
1359:,
1356:y
1345:x
1341:e
1337:=
1331:x
1323:u
1295:.
1292:y
1281:x
1277:e
1273:=
1270:)
1267:y
1264:,
1261:x
1258:(
1255:u
1233:v
1225:u
1213:u
1205:v
1201:u
1187:v
1183:u
1148:0
1145:x
1141:0
1138:x
1136:(
1134:v
1130:v
1114:2
1109:R
1097:u
1068:.
1065:)
1062:y
1059:i
1056:+
1053:x
1050:(
1047:f
1021:u
1001:,
978:)
975:z
972:(
969:f
946:)
943:y
940:,
937:x
934:(
929:y
925:u
921:i
915:)
912:y
909:,
906:x
903:(
898:x
894:u
887:)
884:z
881:(
878:g
858:u
838:.
833:x
830:y
826:u
822:=
817:y
814:x
810:u
785:0
782:=
779:u
754:x
750:u
727:y
723:u
699:,
694:2
689:R
661:u
641:.
614:v
594:u
554:u
534:v
506:u
483:v
463:v
443:u
419:,
416:u
372:.
349:)
346:y
343:,
340:x
337:(
334:v
331:i
328:+
325:)
322:y
319:,
316:x
313:(
310:u
304:)
301:z
298:(
295:f
275:u
255:v
235:.
226:y
223:i
220:+
217:x
211:z
187:)
184:z
181:(
178:f
151:)
148:y
145:,
142:x
139:(
136:v
114:2
109:R
78:)
75:y
72:,
69:x
66:(
63:u
41:.
34:.
20:)
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