325:
9436:
5509:
6065:
10906:
9101:
40:
5211:
104:
5848:
5033:
9711:
9431:{\displaystyle {\begin{aligned}0&=\nabla f=(\sigma _{1}\partial _{x}+\sigma _{2}\partial _{y})(u+\sigma _{1}\sigma _{2}v)\\&=\sigma _{1}\partial _{x}u+\underbrace {\sigma _{1}\sigma _{1}\sigma _{2}} _{=\sigma _{2}}\partial _{x}v+\sigma _{2}\partial _{y}u+\underbrace {\sigma _{2}\sigma _{1}\sigma _{2}} _{=-\sigma _{1}}\partial _{y}v=0\end{aligned}}}
7782:
4226:
7080:
9848:
6060:{\textstyle \nabla u\cdot \nabla v={\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}\cdot {\frac {\partial v}{\partial y}}={\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial x}}\cdot {\frac {\partial v}{\partial x}}=0}
7499:
2841:
9499:
1718:
Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the Cauchy–Riemann equations with a conformal map must itself solve the Cauchy–Riemann equations. Thus the Cauchy–Riemann equations are conformally
8097:
7622:
7251:
2668:
4026:
6932:
4724:, and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.
8633:
5206:{\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right),\;\;\;{\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}
2948:
4571:
4022:
9720:
8355:
7371:
5667:
2673:
2240:
4865:
is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see
3526:
7950:
5783:
8924:
7146:
2523:
7585:
6569:
6478:
4794:
be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function
1165:
1465:
10064:
One might seek to generalize the Cauchy-Riemann equations instead by asking more generally when are the solutions of a system of PDEs closed under composition. The theory of Lie
Pseudogroups addresses these kinds of questions.
8520:
557:
470:
7888:
9030:
1711:. Consequently, a function satisfying the Cauchy–Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane. That is, the Cauchy–Riemann equations are the conditions for a function to be
8221:
obey the Cauchy–Riemann equations throughout the domain Ω is essential. It is possible to construct a continuous function satisfying the Cauchy–Riemann equations at a point, but which is not analytic at the point (e.g.,
3668:
3597:
2848:
9706:{\displaystyle \nabla f=\sigma _{1}(\partial _{x}u-\partial _{y}v)+\sigma _{2}(\partial _{x}v+\partial _{y}u)=0\Leftrightarrow {\begin{cases}\partial _{x}u-\partial _{y}v=0\\\partial _{x}v+\partial _{y}u=0\end{cases}}}
4435:
3870:
7777:{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}&=\alpha (x,y)\\{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}&=\beta (x,y)\end{aligned}}}
3286:
1273:
6917:
6754:
1544:
8242:
5351:
6360:
1667:
1604:
5266:
9106:
7627:
5590:
4221:{\displaystyle {\frac {\Delta f}{\Delta z}}={\frac {f_{x}-if_{y}}{2}}+{\frac {f_{x}+if_{y}}{2}}\cdot {\frac {\Delta {\bar {z}}}{\Delta z}}+{\frac {\eta (\Delta z)}{\Delta z}},\;\;\;\;(\Delta z\neq 0).}
8515:
7075:{\displaystyle Df(x,y)={\begin{bmatrix}{\dfrac {\partial u}{\partial x}}&{\dfrac {\partial u}{\partial y}}\\{\dfrac {\partial v}{\partial x}}&{\dfrac {\partial v}{\partial y}}\end{bmatrix}}}
9984:
3863:
3806:
1064:
3381:
984:
8816:
2112:
9900:
3741:
5694:
4352:
4300:
4982:
7512:
5843:
4658:
1790:
9843:{\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial x}}={\dfrac {\partial v}{\partial y}}\\{\dfrac {\partial u}{\partial y}}=-{\dfrac {\partial v}{\partial x}}\end{cases}}}
4863:
4722:
8460:
7494:{\displaystyle {\partial u \over \partial r}={1 \over r}{\partial v \over \partial \theta },\quad {\partial v \over \partial r}=-{1 \over r}{\partial u \over \partial \theta }.}
10022:
7319:
6585:
one, and it is free from sources or sinks, having net flux equal to zero through any open domain without holes. (These two observations combine as real and imaginary parts in
1987:
7366:
2107:
6499:
6408:
3117:
2461:
6272:
6203:
6175:
6137:
6105:
1059:
8773:
3139:
1402:
1358:
9492:
9465:
6665:
4430:
1315:
9094:
6395:
4381:
3319:
2836:{\displaystyle \lim _{\underset {h\in \mathbb {R} }{h\to 0}}{\frac {f(z_{0}+ih)-f(z_{0})}{ih}}=\left.{\frac {1}{i}}{\frac {\partial f}{\partial y}}\right\vert _{z_{0}}.}
882:
496:
8703:
8960:
8821:
6794:
415:
7817:
6244:
5424:
4929:
2408:
2370:
9065:
8738:
1054:
1019:
840:
4752:
4600:
4251:
3076:
3025:
2325:
1873:
3353:
2975:
2518:
2491:
2267:
5024:
5003:
4900:
4792:
4772:
3049:
2998:
2294:
1830:
8169:
is analytic in an open complex domain Ω if and only if it satisfies the Cauchy–Riemann equation in the domain. In particular, continuous differentiability of
8092:{\displaystyle f\left(\zeta ,{\bar {\zeta }}\right)={\frac {1}{2\pi i}}\iint _{D}\varphi \left(z,{\bar {z}}\right)\,{\frac {dz\wedge d{\bar {z}}}{z-\zeta }}}
5497:
5475:
5449:
5394:
5372:
5287:
2051:
2027:
2007:
1913:
1893:
1810:
1170:
7246:{\displaystyle {\frac {\partial u}{\partial n}}={\frac {\partial v}{\partial s}},\quad {\frac {\partial v}{\partial n}}=-{\frac {\partial u}{\partial s}}}
6825:
6694:
2663:{\displaystyle \lim _{\underset {h\in \mathbb {R} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}=\left.{\frac {\partial f}{\partial x}}\right\vert _{z_{0}}}
1380:. In the theory there are several other major ways of looking at this notion, and the translation of the condition into other language is often needed.
355:
10035:, this system is equivalent to the standard Cauchy–Riemann equations of complex variables, and the solutions are holomorphic functions. In dimension
1490:
8969:
6304:
8237:. Similarly, some additional assumption is needed besides the Cauchy–Riemann equations (such as continuity), as the following example illustrates
5216:
3602:
3531:
8465:
10866:
799:. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.
8628:{\displaystyle {\partial f \over \partial {\bar {z}}}={1 \over 2}\left({\partial f \over \partial x}+i{\partial f \over \partial y}\right).}
5535:
of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.
10659:
9909:
10043:
10686:
3144:
2297:
8422:, where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish.
2943:{\displaystyle i\left.{\frac {\partial f}{\partial x}}\right\vert _{z_{0}}=\left.{\frac {\partial f}{\partial y}}\right\vert _{z_{0}}}
891:
10721:
5293:
1609:
3355:(in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that
1549:
17:
4566:{\displaystyle \left.{\frac {df}{dz}}\right|_{z_{0}}=\lim _{\Delta z\to 0}{\frac {\Delta f}{\Delta z}}={\frac {f_{x}-if_{y}}{2}}.}
4017:{\displaystyle \Delta f(z_{0})={\frac {f_{x}-if_{y}}{2}}\,\Delta z+{\frac {f_{x}+if_{y}}{2}}\,\Delta {\bar {z}}+\eta (\Delta z)\,}
10932:
10726:
6108:
348:
4934:
10285:
10260:
8411:
10493:
Gray, J. D.; Morris, S. A. (April 1978). "When is a
Function that Satisfies the Cauchy–Riemann Equations Analytic?".
5539:
A standard physical interpretation of the Cauchy–Riemann equations going back to
Riemann's work on function theory is that
1730:
1700:
takes infinitesimal line segments at the intersection of two curves in z and rotates them to the corresponding segments in
10180:(1851). "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse". In H. Weber (ed.).
6668:
3811:
1368:
The Cauchy-Riemann equations are one way of looking at the condition for a function to be differentiable in the sense of
8436:
8350:{\displaystyle f(z)={\begin{cases}\exp \left(-z^{-4}\right)&{\text{if }}z\not =0\\0&{\text{if }}z=0\end{cases}}}
3757:
1670:
10841:
10831:
10816:
10736:
10120:
765:
6274:
intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.
10876:
10623:
10583:
10552:
10332:
6283:
341:
206:
8778:
10891:
9864:
5662:{\displaystyle \nabla u={\frac {\partial u}{\partial x}}\mathbf {i} +{\frac {\partial u}{\partial y}}\mathbf {j} .}
389:
10074:
4432:, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case,
3687:
2520:
along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is
10952:
10886:
10881:
10856:
10711:
10679:
10594:
5680:
4305:
1373:
5804:
also satisfies the
Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the
4256:
10604:
393:
10821:
9729:
9619:
8643:
6586:
5532:
4605:
211:
201:
5810:
3521:{\displaystyle \Delta f(z_{0})=f(z_{0}+\Delta z)-f(z_{0})=f_{x}\,\Delta x+f_{y}\,\Delta y+\eta (\Delta z)}
10942:
10846:
10611:
10599:
10350:"Le Potentiel de Vitesse pour les Ecoulements de Fluides Réels: la Contribution de Joseph-Louis Lagrange"
8211:
7922:
4798:
4663:
7589:
The inhomogeneous Cauchy–Riemann equations consist of the two equations for a pair of unknown functions
772:
are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.
10871:
10851:
10104:
8121:
5353:
In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function
4660:, regarded as a complex function with imaginary part identically zero, has both partial derivatives at
2235:{\displaystyle f'(z_{0})=\lim _{\underset {h\in \mathbb {C} }{h\to 0}}{\frac {f(z_{0}+h)-f(z_{0})}{h}}}
788:
9995:
10947:
10937:
10910:
10672:
7332:
7286:
6582:
6398:
1918:
1682:
758:
is a complex function that is differentiable at every point of some open subset of the complex plane
265:
158:
149:
10137:
8266:
5778:{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.}
2417:
2056:
10861:
10751:
10716:
6688:
10047:
8743:
7329:. As a consequence, in particular, in the system of coordinates given by the polar representation
6249:
6180:
6152:
6114:
6082:
3122:
1320:
10771:
9470:
9443:
8919:{\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=1,\sigma _{1}\sigma _{2}+\sigma _{2}\sigma _{1}=0}
7944:
7108:
6630:
6489:
5560:
3378:(real differentiable). This is equivalent to the existence of the following linear approximation
1837:
1280:
726:
187:
9070:
6371:
4386:
3291:
845:
10786:
8676:
5575:
to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the
4357:
3081:
736:
401:
10251:
Arfken, George B.; Weber, Hans J.; Harris, Frank E. (2013). "11.2 CAUCHY-RIEMANN CONDITIONS".
8929:
7580:{\displaystyle {\partial f \over \partial r}={1 \over ir}{\partial f \over \partial \theta }.}
6763:
10811:
10645:
10084:
8415:
8399:
5688:
5027:
2375:
2337:
1377:
381:
281:
256:
83:
9035:
8708:
8647:
8141:
6609:, they model static electric fields in a region of the plane containing no electric charge.
6220:
5400:
4905:
1024:
989:
810:
8651:
8419:
4730:
4578:
3054:
3003:
2303:
1833:
1674:
755:
291:
272:
226:
168:
8398:
This is in fact a special case of a more general result on the regularity of solutions of
4233:
8:
10836:
10746:
10731:
10397:
10079:
7326:
7104:
6140:
178:
140:
10352:[Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange's Contribution].
1843:
775:
This equivalence between differentiability and analyticity is the starting point of all
10756:
10510:
10379:
8366:
Nevertheless, if a function satisfies the Cauchy–Riemann equations in an open set in a
8359:
which satisfies the Cauchy–Riemann equations everywhere, but fails to be continuous at
8191:
7936:
6578:
6574:
6564:{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial (-v)}{\partial y}}=0.}
6473:{\displaystyle {\frac {\partial (-v)}{\partial x}}-{\frac {\partial u}{\partial y}}=0.}
6402:
5544:
5009:
4988:
4885:
4777:
4757:
4727:
Some sources state a sufficient condition for the complex differentiability at a point
3331:
3034:
2983:
2953:
2496:
2469:
2279:
2245:
1815:
796:
329:
236:
8160:
1160:{\displaystyle {\begin{aligned}u(x,y)&=x^{2}-y^{2}\\v(x,y)&=2xy\end{aligned}}}
10826:
10766:
10642:
10619:
10579:
10548:
10383:
10371:
10328:
10281:
10256:
10116:
9903:
8659:
8391:
7120:
6613:
6362:
regarded as a (real) two-component vector. Then the second Cauchy–Riemann equation (
5790:
4879:
4878:
The above proof suggests another interpretation of the Cauchy–Riemann equations. The
1460:{\displaystyle {i{\frac {\partial f}{\partial x}}}={\frac {\partial f}{\partial y}}.}
769:
324:
251:
163:
135:
63:
afterwards. If both of these result in the point ending up in the same place for all
10431:
10761:
10741:
10695:
10502:
10361:
10177:
8671:
6146:
A holomorphic function can therefore be visualized by plotting the two families of
5508:
5482:
5460:
5434:
5379:
5357:
5272:
2273:
2242:
provided this limit exists (that is, the limit exists along every path approaching
2036:
2012:
1992:
1898:
1878:
1795:
1369:
776:
552:{\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}},}
397:
385:
369:
311:
306:
296:
125:
95:
10166:. Oeuvres complètes Ser. 1. Vol. 1. Paris (published 1882). pp. 319–506.
8462:
annihilates holomorphic functions. This generalizes most directly the formulation
7119:
Other representations of the Cauchy–Riemann equations occasionally arise in other
10806:
10781:
10522:
10349:
10159:
10108:
8367:
6922:
5576:
5556:
1484:
465:{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}}
231:
196:
10320:
8410:
There are Cauchy–Riemann equations, appropriately generalized, in the theory of
7883:{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=\varphi (z,{\bar {z}})}
10796:
10791:
10401:
10133:
9025:{\displaystyle \nabla \equiv \sigma _{1}\partial _{x}+\sigma _{2}\partial _{y}}
8963:
7927:, then the inhomogeneous equation is explicitly solvable in any bounded domain
6672:
6606:
6602:
6598:
6594:
6590:
5548:
2411:
2328:
792:
633:
629:
606:
286:
221:
216:
111:
1483:
In this form, the equations correspond structurally to the condition that the
10926:
10375:
8650:. More complicated, generally non-linear Bäcklund transforms, such as in the
3663:{\textstyle f_{y}=\left.{\frac {\partial f}{\partial y}}\right\vert _{z_{0}}}
3592:{\textstyle f_{x}=\left.{\frac {\partial f}{\partial x}}\right\vert _{z_{0}}}
2332:
1712:
1389:
246:
241:
130:
4754:
as, in addition to the Cauchy–Riemann equations, the partial derivatives of
10776:
10571:
10540:
10046:
implies, under suitable smoothness assumptions, that any such mapping is a
6676:
6299:
5512:
10366:
10801:
10302:
10059:
7322:
6147:
5805:
373:
301:
120:
33:
8206:
exist in Ω, and satisfy the Cauchy–Riemann equations throughout Ω, then
10514:
8176:
The hypotheses of
Goursat's theorem can be weakened significantly. If
6493:
6282:
Another interpretation of the Cauchy–Riemann equations can be found in
6214:
5794:
5213:
the Cauchy–Riemann equations can then be written as a single equation
4602:
is essential and cannot be dispensed with. For example, the function
1268:{\displaystyle u_{x}=2x;\quad u_{y}=-2y;\quad v_{x}=2y;\quad v_{y}=2x}
10650:
7092:
satisfies the Cauchy–Riemann equations if and only if the 2×2 matrix
4354:. Therefore, the second term is independent of the path of the limit
1989:
so the function can also be regarded as a function of real variables
625:
10506:
8418:
of PDEs. This is done using a straightforward generalization of the
6912:{\displaystyle f(x,y)={\begin{bmatrix}u(x,y)\\v(x,y)\end{bmatrix}}.}
6612:
This interpretation can equivalently be restated in the language of
10664:
10309:. Translated by Frances Hardcastle. Cambridge: MacMillan and Bowes.
6625:
5580:
3028:
1678:
1394:
First, the Cauchy–Riemann equations may be written in complex form
32:"Cauchy–Riemann" redirects here. For Cauchy–Riemann manifolds, see
6749:{\displaystyle J={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}.}
5671:
By differentiating the Cauchy–Riemann equations for the functions
4867:
10640:
8655:
6687:
Another formulation of the Cauchy–Riemann equations involves the
6246:, the stationary points of the flow, the equipotential curves of
3281:{\displaystyle |f(z)-f(z_{0})-f'(z_{0})(z-z_{0})|/|z-z_{0}|\to 0}
39:
10042:, this is still sometimes called the Cauchy–Riemann system, and
6075:
are orthogonal to each other. This implies that the gradient of
4575:
Note that the hypothesis of real differentiability at the point
1539:{\displaystyle {\begin{pmatrix}a&-b\\b&a\end{pmatrix}},}
10454:
5426:. As such, we can view analytic functions as true functions of
5346:{\textstyle {\frac {df}{dz}}={\frac {\partial f}{\partial z}}.}
1277:
We see that indeed the Cauchy–Riemann equations are satisfied,
103:
10307:
On
Riemann's theory of algebraic functions and their integrals
10278:
Mathematical
Physics: A Modern Introduction to Its Foundations
7814:. These equations are usually combined into a single equation
6965:
6355:{\displaystyle {\bar {f}}={\begin{bmatrix}u\\-v\end{bmatrix}}}
5527: = const, red) are perpendicular to equipotentials (
1662:{\displaystyle b=\partial v/\partial x=-\partial u/\partial y}
10412:
5797:
of the gradient is zero, and so the fluid is incompressible.
1686:
1599:{\displaystyle a=\partial u/\partial x=\partial v/\partial y}
739:
at a complex point if and only if the partial derivatives of
10390:
8115:
6756:
This is a complex structure in the sense that the square of
5261:{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0,}
10138:"Ulterior disquisitio de formulis integralibus imaginariis"
9836:
9699:
8343:
4441:
3621:
3550:
2901:
2857:
2779:
2621:
787:
The Cauchy–Riemann equations first appeared in the work of
10253:
Mathematical
Methods for Physicists: A Comprehensive Guide
9852:
8386:, and satisfies the Cauchy–Riemann equations weakly, then
6294:
satisfy the Cauchy–Riemann equations in an open subset of
10200:
10142:
Nova Acta
Academiae Scientiarum Imperialis Petropolitanae
10110:
Essai d'une nouvelle théorie de la résistance des fluides
8646:, the Cauchy–Riemann equations are a simple example of a
8510:{\displaystyle {\partial f \over \partial {\bar {z}}}=0,}
2950:
which is the complex form of Cauchy–Riemann equations at
10424:
8382:) is locally integrable in an open domain Ω ⊂
8210:
is holomorphic (and thus analytic). This result is the
6624:
satisfy the Cauchy–Riemann equations if and only if the
5579:
of the fluid at each point of the plane is equal to the
4873:
2463:
and satisfy the Cauchy–Riemann equations at this point.
4870:), this distinction is often elided in the literature.
9861:. The equation for an orientation-preserving mapping
7289:
6855:
6709:
6328:
6252:
6223:
6183:
6155:
6117:
6085:
5851:
5813:
5523:
satisfying the Cauchy–Riemann equations. Streamlines (
5485:
5463:
5437:
5403:
5382:
5360:
5296:
5275:
4908:
4389:
4360:
4308:
4259:
4236:
3814:
3760:
3690:
3605:
3534:
3334:
2956:
2499:
2472:
2248:
2059:
2039:
2015:
1995:
1921:
1901:
1881:
1846:
1798:
1499:
396:
which form a necessary and sufficient condition for a
10442:
9998:
9912:
9867:
9812:
9784:
9758:
9733:
9723:
9502:
9473:
9446:
9104:
9073:
9038:
8972:
8932:
8824:
8781:
8746:
8711:
8679:
8523:
8468:
8439:
8245:
7953:
7820:
7625:
7515:
7374:
7335:
7149:
7043:
7019:
6993:
6969:
6935:
6828:
6766:
6697:
6633:
6502:
6411:
6374:
6307:
5697:
5593:
5571:
satisfies the Cauchy–Riemann equations. We will take
5219:
5036:
5012:
4991:
4937:
4888:
4801:
4780:
4760:
4733:
4666:
4608:
4581:
4438:
4029:
3873:
3384:
3294:
3147:
3125:
3084:
3057:
3037:
3006:
2986:
2851:
2676:
2526:
2420:
2378:
2340:
2306:
2282:
2115:
1818:
1733:
1612:
1552:
1493:
1405:
1323:
1283:
1173:
1062:
1027:
992:
894:
848:
813:
499:
418:
10466:
9979:{\displaystyle Df^{\mathsf {T}}Df=(\det(Df))^{2/n}I}
6682:
3858:{\textstyle \Delta z-\Delta {\bar {z}}=2i\,\Delta y}
751:
satisfy the Cauchy–Riemann equations at that point.
10184:(in German). Dover (published 1953). pp. 3–48.
6605:on a region of the plane containing no current. In
3801:{\textstyle \Delta z+\Delta {\bar {z}}=2\,\Delta x}
1363:
10660:Cauchy–Riemann Equations Module by John H. Mathews
10276:Hassani, Sadri (2013). "10.2 Analytic Functions".
10255:(7th ed.). Academic Press. pp. 471–472.
10016:
9978:
9894:
9842:
9705:
9486:
9459:
9430:
9088:
9059:
9024:
8954:
8918:
8810:
8767:
8732:
8697:
8627:
8509:
8454:
8370:, then the function is analytic. More precisely:
8349:
8091:
7882:
7776:
7579:
7493:
7360:
7313:
7245:
7107:, where it is the starting point for the study of
7074:
6911:
6788:
6748:
6659:
6563:
6472:
6389:
6354:
6266:
6238:
6197:
6169:
6131:
6099:
6059:
5837:
5777:
5661:
5491:
5469:
5443:
5418:
5388:
5366:
5345:
5281:
5260:
5205:
5018:
4997:
4976:
4923:
4894:
4857:
4786:
4766:
4746:
4716:
4652:
4594:
4565:
4424:
4375:
4346:
4294:
4245:
4220:
4016:
3857:
3800:
3735:
3662:
3591:
3520:
3347:
3313:
3280:
3133:
3111:
3070:
3043:
3019:
2992:
2969:
2942:
2835:
2662:
2512:
2485:
2455:
2402:
2364:
2319:
2288:
2261:
2234:
2101:
2045:
2021:
2001:
1981:
1907:
1887:
1867:
1824:
1804:
1784:
1661:
1598:
1538:
1459:
1352:
1309:
1267:
1159:
1048:
1013:
978:
876:
834:
551:
464:
10475:Geometric function theory and non-linear analysis
10250:
8665:
2493:, then it may be computed by taking the limit at
1372:: in other words, they encapsulate the notion of
47:in a domain being multiplied by a complex number
10924:
10188:
9940:
9096:, which can be calculated in the following way:
5531: = const, blue). The point (0,0) is a
4484:
2678:
2528:
2144:
979:{\displaystyle f(z)=(x+iy)^{2}=x^{2}-y^{2}+2ixy}
10396:
6067:), i.e., the direction of the maximum slope of
10578:(3rd ed.). McGraw Hill (published 1979).
10547:(3rd ed.). McGraw Hill (published 1987).
10530:
10406:Foundations of differential geometry, volume 2
10313:
10206:
10103:
8811:{\displaystyle J\equiv \sigma _{1}\sigma _{2}}
7135:) hold for a differentiable pair of functions
4383:when (and only when) it vanishes identically:
27:Chacteristic property of holomorphic functions
10680:
10472:
10294:
9895:{\displaystyle f:\Omega \to \mathbb {R} ^{n}}
8425:
2845:So, the equality of the derivatives implies
2466:In fact, if the complex derivative exists at
1673:. Geometrically, such a matrix is always the
349:
10722:Grothendieck–Hirzebruch–Riemann–Roch theorem
10280:(2nd ed.). Springer. pp. 300–301.
10218:
9857:Let Ω be an open set in the Euclidean space
6760:is the negative of the 2×2 identity matrix:
3736:{\textstyle \eta (\Delta z)/|\Delta z|\to 0}
1722:
10610:
10592:
10341:
10221:Theory of functions of a complex variable 1
10097:
2670:and along the imaginary axis, the limit is
2269:, and does not depend on the chosen path).
10687:
10673:
10492:
10460:
10319:
10234:
10170:
5503:
5116:
5115:
5114:
4347:{\textstyle \Delta {\bar {z}}/\Delta z=-1}
4196:
4195:
4194:
4193:
356:
342:
10867:Riemann–Roch theorem for smooth manifolds
10430:
10365:
10152:
9882:
8654:, are of great interest in the theory of
8116:Goursat's theorem and its generalizations
8047:
6650:
6637:
4295:{\textstyle \Delta {\bar {z}}/\Delta z=1}
4013:
3979:
3934:
3848:
3791:
3493:
3473:
3127:
3027:, it is also real differentiable and the
2702:
2552:
2168:
1671:matrix representation of a complex number
10126:
7114:
6277:
5507:
4977:{\displaystyle {\overline {x+iy}}:=x-iy}
2327:(that is, it has a complex-derivative),
38:
10570:
10347:
10275:
10176:
9992:is the Jacobian matrix, with transpose
9853:Conformal mappings in higher dimensions
8966:in this Clifford algebra is defined as
8190:is continuous in an open set Ω and the
75:satisfies the Cauchy–Riemann condition.
14:
10925:
10618:(1st ed.). CUP (published 1984).
10521:
10448:
10225:, p. 110-112 (Translated from Russian)
10158:
10008:
9922:
9067:is considered analytic if and only if
8140:is a complex-valued function which is
7503:Combining these into one equation for
6820:) are two functions in the plane, put
6217:family of curves. At the points where
6213:) is not zero, these families form an
10668:
10641:
10539:
10418:
10301:
10194:
10132:
8637:
6929:is the matrix of partial derivatives
5838:{\textstyle \nabla u\cdot \nabla v=0}
4874:Independence of the complex conjugate
4653:{\displaystyle f(x,y)={\sqrt {|xy|}}}
1785:{\displaystyle f(z)=u(z)+i\cdot v(z)}
1383:
10694:
9906:(that is, angle-preserving) is that
8405:
6205:. Near points where the gradient of
3359:as a function of two real variables
1396:
490:
409:
10325:Problems and theorems in analysis I
10164:Mémoire sur les intégrales définies
10115:Reprint 2018 by Hachette Livre-BNF
10053:
7368:, the equations then take the form
6488:) asserts that the vector field is
6482:The first Cauchy–Riemann equation (
4858:{\displaystyle f(z)=z^{2}e^{i/|z|}}
4717:{\displaystyle (x_{0},y_{0})=(0,0)}
3119:, regarded as a real-linear map of
24:
10832:Riemannian connection on a surface
10737:Measurable Riemann mapping theorem
10563:
10028:denotes the identity matrix. For
9874:
9823:
9815:
9795:
9787:
9769:
9761:
9744:
9736:
9678:
9662:
9639:
9623:
9590:
9574:
9542:
9526:
9503:
9406:
9331:
9305:
9233:
9165:
9142:
9119:
9074:
9013:
8990:
8973:
8608:
8600:
8582:
8574:
8535:
8527:
8480:
8472:
8455:{\displaystyle {\bar {\partial }}}
8443:
8110:
7832:
7824:
7736:
7728:
7713:
7705:
7664:
7656:
7641:
7633:
7565:
7557:
7527:
7519:
7479:
7471:
7443:
7435:
7419:
7411:
7386:
7378:
7302:
7293:
7234:
7226:
7208:
7200:
7184:
7176:
7161:
7153:
7054:
7046:
7030:
7022:
7004:
6996:
6980:
6972:
6601:, such vector fields model static
6546:
6529:
6514:
6506:
6455:
6447:
6432:
6415:
6224:
6042:
6034:
6019:
6011:
5996:
5988:
5973:
5965:
5950:
5942:
5927:
5919:
5904:
5896:
5881:
5873:
5861:
5852:
5823:
5814:
5753:
5739:
5716:
5702:
5642:
5634:
5614:
5606:
5594:
5559:. Suppose that the pair of (twice
5452:) instead of complex functions of
5331:
5323:
5231:
5223:
5186:
5182:
5165:
5161:
5123:
5119:
5097:
5093:
5076:
5072:
5043:
5039:
4513:
4505:
4488:
4361:
4329:
4309:
4280:
4260:
4237:
4200:
4181:
4170:
4149:
4132:
4041:
4033:
4004:
3980:
3935:
3874:
3849:
3824:
3815:
3792:
3770:
3761:
3716:
3697:
3634:
3626:
3563:
3555:
3509:
3494:
3474:
3429:
3385:
2914:
2906:
2870:
2862:
2803:
2795:
2634:
2626:
1653:
1642:
1630:
1619:
1590:
1579:
1570:
1559:
1445:
1437:
1421:
1413:
1167:and their partial derivatives are
766:holomorphic functions are analytic
537:
529:
511:
503:
453:
445:
430:
422:
25:
10964:
10634:
10495:The American Mathematical Monthly
7103:This interpretation is useful in
6683:Preservation of complex structure
4302:, while if it is imaginary, then
3865:, the above can be re-written as
802:
10905:
10904:
10473:Iwaniec, T.; Martin, G. (2001).
10182:Riemann's gesammelte math. Werke
10017:{\displaystyle Df^{\mathsf {T}}}
9715:Hence, in traditional notation:
8402:partial differential equations.
7314:{\textstyle (\nabla n,\nabla s)}
6581:, such a field is necessarily a
5652:
5624:
1982:{\textstyle f(z)=f(x+iy)=f(x,y)}
1364:Interpretation and reformulation
323:
102:
10817:Riemann's differential equation
10727:Hirzebruch–Riemann–Roch theorem
10531:Marsden, A; Hoffman, M (1973).
10438:. Academic Press. §9.10, Ex. 1.
10269:
10075:List of complex analysis topics
8394:with an analytic function in Ω.
7431:
7361:{\displaystyle z=re^{i\theta }}
7196:
5397:is independent of the variable
2102:{\textstyle z_{0}=x_{0}+iy_{0}}
1669:. A matrix of this form is the
1245:
1222:
1196:
884:is differentiable at any point
43:A visual depiction of a vector
10933:Partial differential equations
10842:Riemann–Hilbert correspondence
10712:Generalized Riemann hypothesis
10436:Foundations of modern analysis
10244:
10228:
10212:
9956:
9952:
9943:
9937:
9877:
9611:
9602:
9570:
9554:
9522:
9209:
9177:
9174:
9128:
8692:
8686:
8666:Definition in Clifford algebra
8544:
8489:
8446:
8255:
8249:
8069:
8036:
7974:
7877:
7871:
7856:
7841:
7767:
7755:
7695:
7683:
7308:
7290:
6954:
6942:
6895:
6883:
6873:
6861:
6844:
6832:
6541:
6532:
6427:
6418:
6381:
6314:
5681:symmetry of second derivatives
5410:
5268:and the complex derivative of
5240:
5132:
4915:
4849:
4841:
4811:
4805:
4711:
4699:
4693:
4667:
4644:
4633:
4624:
4612:
4494:
4367:
4318:
4269:
4212:
4197:
4176:
4167:
4141:
4010:
4001:
3989:
3893:
3880:
3833:
3779:
3727:
3723:
3712:
3703:
3694:
3515:
3506:
3457:
3444:
3435:
3413:
3404:
3391:
3298:
3272:
3268:
3247:
3237:
3233:
3214:
3211:
3198:
3184:
3171:
3162:
3156:
3149:
3106:
3093:
2760:
2747:
2738:
2716:
2687:
2607:
2594:
2585:
2566:
2537:
2456:{\displaystyle (x_{0},y_{0}),}
2447:
2421:
2397:
2382:
2359:
2344:
2223:
2210:
2201:
2182:
2153:
2137:
2124:
1976:
1964:
1955:
1940:
1931:
1925:
1779:
1773:
1758:
1752:
1743:
1737:
1685:, and in particular preserves
1374:function of a complex variable
1134:
1122:
1082:
1070:
1043:
1031:
1008:
996:
926:
910:
904:
898:
858:
852:
842:. The complex-valued function
394:partial differential equations
13:
1:
10877:Riemann–Siegel theta function
10090:
7810:defined in an open subset of
6267:{\textstyle u={\text{const}}}
6198:{\textstyle v={\text{const}}}
6170:{\textstyle u={\text{const}}}
6132:{\textstyle u={\text{const}}}
6100:{\textstyle v={\text{const}}}
3000:is complex differentiable at
1689:. The Jacobian of a function
795:connected this system to the
729:of the real variables. Then
687:of a single complex variable
10892:Riemann–von Mangoldt formula
10408:. Wiley. Proposition IX.2.2.
8768:{\displaystyle z\equiv x+Jy}
8644:conjugate harmonic functions
7131:
7125:
6484:
6364:
4954:
3134:{\displaystyle \mathbb {C} }
1353:{\displaystyle u_{y}=-v_{x}}
565:
478:
400:of a complex variable to be
7:
10600:Encyclopedia of Mathematics
10595:"Cauchy–Riemann conditions"
10593:Solomentsev, E.D. (2001) ,
10219:Markushevich, A.I. (1977).
10068:
9487:{\displaystyle \sigma _{2}}
9460:{\displaystyle \sigma _{1}}
7084:Then the pair of functions
6660:{\displaystyle v\,dx+u\,dy}
6593:, such a vector field is a
5561:continuously differentiable
4425:{\textstyle f_{x}+if_{y}=0}
1473:
1310:{\displaystyle u_{x}=v_{y}}
10:
10969:
10887:Riemann–Stieltjes integral
10882:Riemann–Silberstein vector
10857:Riemann–Liouville integral
10646:"Cauchy–Riemann Equations"
10485:
10354:Journal la Houille Blanche
10239:. Oxford University Press.
10207:Marsden & Hoffman 1973
10057:
9089:{\displaystyle \nabla f=0}
8426:Complex differential forms
8414:. They form a significant
8119:
7255:for any coordinate system
6677:harmonic differential form
6390:{\displaystyle {\bar {f}}}
4376:{\textstyle \Delta z\to 0}
3314:{\displaystyle z\to z_{0}}
1387:
877:{\displaystyle f(z)=z^{2}}
782:
770:analytic complex functions
764:. It has been proved that
31:
10900:
10822:Riemann's minimal surface
10702:
10545:Real and complex analysis
10432:Dieudonné, Jean Alexandre
8698:{\displaystyle C\ell (2)}
8430:As often formulated, the
8412:several complex variables
7786:for some given functions
6587:Cauchy's integral theorem
6107:curves; so these are the
3112:{\displaystyle f'(z_{0})}
1723:Complex differentiability
1376:by means of conventional
266:Geometric function theory
212:Cauchy's integral formula
202:Cauchy's integral theorem
59:then being multiplied by
55:, versus being mapped by
10847:Riemann–Hilbert problems
10752:Riemann curvature tensor
10717:Grand Riemann hypothesis
10707:Cauchy–Riemann equations
8955:{\displaystyle J^{2}=-1}
7109:pseudoholomorphic curves
6789:{\displaystyle J^{2}=-I}
2272:A fundamental result of
727:differentiable functions
378:Cauchy–Riemann equations
174:Cauchy–Riemann equations
18:Cauchy-Riemann equations
10772:Riemann mapping theorem
10323:; Szegő, Gábor (1978).
10237:The theory of functions
8212:Looman–Menchoff theorem
7945:Cauchy integral formula
6691:in the plane, given by
6239:{\textstyle \nabla u=0}
5504:Physical interpretation
5419:{\textstyle {\bar {z}}}
4924:{\textstyle {\bar {z}}}
2403:{\displaystyle v(x+iy)}
2365:{\displaystyle u(x+iy)}
1838:complex-valued function
1021:and the imaginary part
888:in the complex plane.
789:Jean le Rond d'Alembert
159:Complex-valued function
10872:Riemann–Siegel formula
10852:Riemann–Lebesgue lemma
10787:Riemann series theorem
10614:; Tall, David (1983).
10533:Basic complex analysis
10461:Gray & Morris 1978
10235:Titchmarsh, E (1939).
10113:. Paris: David l'aîné.
10018:
9980:
9896:
9844:
9707:
9488:
9461:
9432:
9090:
9061:
9060:{\displaystyle f=u+Jv}
9026:
8956:
8920:
8812:
8769:
8734:
8733:{\displaystyle z=x+iy}
8699:
8629:
8511:
8456:
8351:
8122:Cauchy–Goursat theorem
8093:
7884:
7778:
7619:of two real variables
7581:
7495:
7362:
7315:
7247:
7076:
6913:
6790:
6750:
6661:
6573:Owing respectively to
6565:
6474:
6391:
6356:
6268:
6240:
6199:
6171:
6133:
6101:
6061:
5839:
5793:. This means that the
5779:
5663:
5536:
5493:
5471:
5445:
5420:
5390:
5375:of a complex variable
5368:
5347:
5283:
5262:
5207:
5020:
4999:
4978:
4925:
4896:
4859:
4788:
4768:
4748:
4718:
4654:
4596:
4567:
4426:
4377:
4348:
4296:
4247:
4222:
4018:
3859:
3802:
3737:
3664:
3593:
3522:
3349:
3315:
3282:
3135:
3113:
3078:is the complex scalar
3072:
3045:
3021:
2994:
2971:
2944:
2837:
2664:
2514:
2487:
2457:
2404:
2366:
2321:
2298:complex differentiable
2290:
2263:
2236:
2103:
2047:
2023:
2003:
1983:
1909:
1889:
1869:
1840:of a complex variable
1826:
1806:
1786:
1663:
1600:
1540:
1461:
1354:
1311:
1269:
1161:
1050:
1049:{\displaystyle v(x,y)}
1015:
1014:{\displaystyle u(x,y)}
980:
878:
836:
835:{\displaystyle z=x+iy}
737:complex differentiable
553:
466:
402:complex differentiable
330:Mathematics portal
76:
10953:Augustin-Louis Cauchy
10812:Riemann zeta function
10524:Göttinger Nachrichten
10477:. Oxford. p. 32.
10085:Wirtinger derivatives
10048:Möbius transformation
10019:
9981:
9897:
9845:
9708:
9489:
9462:
9433:
9091:
9062:
9027:
8957:
8921:
8813:
8770:
8735:
8705:, the complex number
8700:
8630:
8512:
8457:
8416:overdetermined system
8352:
8173:need not be assumed.
8094:
7935:is continuous on the
7885:
7779:
7582:
7496:
7363:
7316:
7248:
7115:Other representations
7077:
6914:
6791:
6751:
6662:
6566:
6475:
6392:
6357:
6278:Harmonic vector field
6269:
6241:
6200:
6172:
6134:
6102:
6079:must point along the
6062:
5840:
5780:
5664:
5547:of an incompressible
5511:
5494:
5472:
5446:
5421:
5391:
5369:
5348:
5284:
5263:
5208:
5028:Wirtinger derivatives
5021:
5000:
4979:
4926:
4897:
4860:
4789:
4769:
4749:
4747:{\displaystyle z_{0}}
4719:
4655:
4597:
4595:{\displaystyle z_{0}}
4568:
4427:
4378:
4349:
4297:
4248:
4246:{\textstyle \Delta z}
4223:
4019:
3860:
3803:
3738:
3665:
3594:
3523:
3369:is differentiable at
3350:
3328:is differentiable at
3316:
3283:
3136:
3114:
3073:
3071:{\displaystyle z_{0}}
3046:
3022:
3020:{\displaystyle z_{0}}
2995:
2972:
2945:
2838:
2665:
2515:
2488:
2458:
2405:
2367:
2322:
2320:{\displaystyle z_{0}}
2291:
2264:
2237:
2104:
2048:
2024:
2004:
1984:
1910:
1890:
1870:
1834:real-valued functions
1827:
1807:
1787:
1664:
1601:
1541:
1462:
1388:Further information:
1378:differential calculus
1355:
1312:
1270:
1162:
1051:
1016:
981:
879:
837:
624:are respectively the
609:bivariate functions.
554:
467:
407:These equations are
282:Augustin-Louis Cauchy
84:Mathematical analysis
42:
10862:Riemann–Roch theorem
10526:(in German): 97–108.
10398:Kobayashi, Shoshichi
10348:Chanson, H. (2007).
9996:
9910:
9865:
9721:
9500:
9471:
9444:
9102:
9071:
9036:
8970:
8930:
8822:
8779:
8744:
8709:
8677:
8652:sine-Gordon equation
8521:
8466:
8437:
8420:Wirtinger derivative
8243:
8217:The hypothesis that
7951:
7818:
7623:
7513:
7372:
7333:
7287:
7147:
6933:
6826:
6764:
6695:
6631:
6500:
6409:
6372:
6305:
6250:
6221:
6181:
6153:
6141:equipotential curves
6115:
6083:
5849:
5811:
5695:
5591:
5483:
5461:
5435:
5401:
5380:
5358:
5294:
5273:
5217:
5034:
5010:
4989:
4935:
4906:
4886:
4799:
4778:
4758:
4731:
4664:
4606:
4579:
4436:
4387:
4358:
4306:
4257:
4234:
4027:
3871:
3812:
3758:
3688:
3603:
3532:
3382:
3332:
3292:
3145:
3123:
3082:
3055:
3035:
3004:
2984:
2954:
2849:
2674:
2524:
2497:
2470:
2418:
2376:
2338:
2304:
2280:
2246:
2113:
2057:
2037:
2013:
1993:
1919:
1915:are real variables.
1899:
1879:
1844:
1816:
1796:
1731:
1610:
1550:
1491:
1403:
1321:
1281:
1171:
1060:
1025:
990:
892:
846:
811:
756:holomorphic function
713:are real variables;
497:
416:
292:Carl Friedrich Gauss
227:Isolated singularity
169:Holomorphic function
10837:Riemannian geometry
10747:Riemann Xi function
10732:Local zeta function
10367:10.1051/lhb:2007072
10160:Cauchy, Augustin L.
10044:Liouville's theorem
8857:
8839:
8192:partial derivatives
7327:positively oriented
7283:such that the pair
7105:symplectic geometry
6298:, and consider the
6209:(or, equivalently,
5026:. Defining the two
4984:for real variables
1868:{\textstyle z=x+iy}
179:Formal power series
141:Unit complex number
10943:Harmonic functions
10757:Riemann hypothesis
10643:Weisstein, Eric W.
10014:
9976:
9892:
9840:
9835:
9831:
9803:
9777:
9752:
9703:
9698:
9484:
9457:
9428:
9426:
9404:
9384:
9303:
9286:
9086:
9057:
9022:
8952:
8916:
8843:
8825:
8808:
8765:
8740:is represented as
8730:
8695:
8660:integrable systems
8648:Bäcklund transform
8638:Bäcklund transform
8625:
8507:
8452:
8347:
8342:
8089:
7943:. Indeed, by the
7880:
7774:
7772:
7577:
7491:
7358:
7311:
7243:
7121:coordinate systems
7072:
7066:
7062:
7038:
7012:
6988:
6909:
6900:
6786:
6746:
6737:
6657:
6614:differential forms
6579:divergence theorem
6561:
6470:
6387:
6352:
6346:
6264:
6236:
6195:
6167:
6129:
6097:
6057:
5835:
5775:
5689:Laplace's equation
5659:
5551:in the plane, and
5545:velocity potential
5537:
5489:
5467:
5441:
5430:complex variable (
5416:
5386:
5364:
5343:
5279:
5258:
5203:
5016:
4995:
4974:
4921:
4892:
4855:
4784:
4764:
4744:
4714:
4650:
4592:
4563:
4501:
4422:
4373:
4344:
4292:
4243:
4218:
4014:
3855:
3798:
3733:
3660:
3589:
3518:
3348:{\textstyle z_{0}}
3345:
3311:
3278:
3141:, since the limit
3131:
3109:
3068:
3041:
3017:
2990:
2970:{\textstyle z_{0}}
2967:
2940:
2833:
2709:
2707:
2660:
2559:
2557:
2513:{\textstyle z_{0}}
2510:
2486:{\textstyle z_{0}}
2483:
2453:
2400:
2362:
2317:
2286:
2262:{\textstyle z_{0}}
2259:
2232:
2175:
2173:
2099:
2043:
2031:complex-derivative
2019:
1999:
1979:
1905:
1885:
1865:
1822:
1802:
1782:
1659:
1596:
1536:
1527:
1457:
1384:Conformal mappings
1350:
1307:
1265:
1157:
1155:
1046:
1011:
976:
874:
832:
797:analytic functions
549:
462:
257:Laplace's equation
237:Argument principle
77:
10920:
10919:
10827:Riemannian circle
10767:Riemann invariant
10287:978-3-319-01195-0
10262:978-0-12-384654-9
10178:Riemann, Bernhard
9904:conformal mapping
9830:
9802:
9776:
9751:
9348:
9346:
9250:
9248:
8615:
8589:
8564:
8551:
8547:
8496:
8492:
8449:
8406:Several variables
8392:almost everywhere
8329:
8306:
8087:
8072:
8039:
8004:
7977:
7874:
7848:
7844:
7743:
7720:
7671:
7648:
7572:
7552:
7534:
7486:
7466:
7450:
7426:
7406:
7393:
7241:
7215:
7191:
7168:
7061:
7037:
7011:
6987:
6689:complex structure
6553:
6521:
6462:
6439:
6384:
6317:
6284:Pólya & Szegő
6262:
6193:
6165:
6127:
6111:of the flow. The
6095:
6049:
6026:
6003:
5980:
5957:
5934:
5911:
5888:
5791:harmonic function
5767:
5730:
5683:, one shows that
5649:
5621:
5549:steady fluid flow
5413:
5338:
5315:
5247:
5243:
5193:
5172:
5152:
5139:
5135:
5104:
5083:
5063:
5050:
5019:{\displaystyle y}
4998:{\displaystyle x}
4957:
4918:
4895:{\displaystyle z}
4880:complex conjugate
4787:{\displaystyle v}
4767:{\displaystyle u}
4648:
4558:
4520:
4483:
4461:
4321:
4272:
4188:
4156:
4144:
4124:
4086:
4048:
3992:
3977:
3932:
3836:
3782:
3641:
3570:
3044:{\displaystyle f}
2993:{\displaystyle f}
2921:
2877:
2810:
2790:
2772:
2682:
2677:
2641:
2614:
2532:
2527:
2289:{\displaystyle f}
2230:
2148:
2143:
1825:{\displaystyle v}
1481:
1480:
1452:
1428:
636:-valued function
573:
572:
544:
518:
486:
485:
460:
437:
366:
365:
252:Harmonic function
164:Analytic function
150:Complex functions
136:Complex conjugate
51:, then mapped by
16:(Redirected from
10960:
10948:Bernhard Riemann
10938:Complex analysis
10908:
10907:
10762:Riemann integral
10742:Riemann (crater)
10696:Bernhard Riemann
10689:
10682:
10675:
10666:
10665:
10656:
10655:
10629:
10616:Complex Analysis
10607:
10589:
10576:Complex analysis
10558:
10536:
10535:. W. H. Freeman.
10527:
10518:
10479:
10478:
10470:
10464:
10458:
10452:
10446:
10440:
10439:
10428:
10422:
10416:
10410:
10409:
10394:
10388:
10387:
10369:
10345:
10339:
10338:
10317:
10311:
10310:
10298:
10292:
10291:
10273:
10267:
10266:
10248:
10242:
10240:
10232:
10226:
10224:
10216:
10210:
10204:
10198:
10192:
10186:
10185:
10174:
10168:
10167:
10156:
10150:
10149:
10130:
10124:
10114:
10105:d'Alembert, Jean
10101:
10080:Morera's theorem
10054:Lie Pseudogroups
10041:
10034:
10023:
10021:
10020:
10015:
10013:
10012:
10011:
9985:
9983:
9982:
9977:
9972:
9971:
9967:
9927:
9926:
9925:
9901:
9899:
9898:
9893:
9891:
9890:
9885:
9849:
9847:
9846:
9841:
9839:
9838:
9832:
9829:
9821:
9813:
9804:
9801:
9793:
9785:
9778:
9775:
9767:
9759:
9753:
9750:
9742:
9734:
9712:
9710:
9709:
9704:
9702:
9701:
9686:
9685:
9670:
9669:
9647:
9646:
9631:
9630:
9598:
9597:
9582:
9581:
9569:
9568:
9550:
9549:
9534:
9533:
9521:
9520:
9493:
9491:
9490:
9485:
9483:
9482:
9466:
9464:
9463:
9458:
9456:
9455:
9437:
9435:
9434:
9429:
9427:
9414:
9413:
9403:
9402:
9401:
9385:
9380:
9379:
9378:
9369:
9368:
9359:
9358:
9339:
9338:
9329:
9328:
9313:
9312:
9302:
9301:
9300:
9287:
9282:
9281:
9280:
9271:
9270:
9261:
9260:
9241:
9240:
9231:
9230:
9215:
9205:
9204:
9195:
9194:
9173:
9172:
9163:
9162:
9150:
9149:
9140:
9139:
9095:
9093:
9092:
9087:
9066:
9064:
9063:
9058:
9031:
9029:
9028:
9023:
9021:
9020:
9011:
9010:
8998:
8997:
8988:
8987:
8961:
8959:
8958:
8953:
8942:
8941:
8925:
8923:
8922:
8917:
8909:
8908:
8899:
8898:
8886:
8885:
8876:
8875:
8856:
8851:
8838:
8833:
8817:
8815:
8814:
8809:
8807:
8806:
8797:
8796:
8774:
8772:
8771:
8766:
8739:
8737:
8736:
8731:
8704:
8702:
8701:
8696:
8672:Clifford algebra
8634:
8632:
8631:
8626:
8621:
8617:
8616:
8614:
8606:
8598:
8590:
8588:
8580:
8572:
8565:
8557:
8552:
8550:
8549:
8548:
8540:
8533:
8525:
8516:
8514:
8513:
8508:
8497:
8495:
8494:
8493:
8485:
8478:
8470:
8461:
8459:
8458:
8453:
8451:
8450:
8442:
8363: = 0.
8356:
8354:
8353:
8348:
8346:
8345:
8330:
8327:
8307:
8304:
8300:
8296:
8295:
8294:
8236:
8198:with respect to
8189:
8157:
8139:
8098:
8096:
8095:
8090:
8088:
8086:
8075:
8074:
8073:
8065:
8049:
8046:
8042:
8041:
8040:
8032:
8015:
8014:
8005:
8003:
7989:
7984:
7980:
7979:
7978:
7970:
7889:
7887:
7886:
7881:
7876:
7875:
7867:
7849:
7847:
7846:
7845:
7837:
7830:
7822:
7809:
7797:
7783:
7781:
7780:
7775:
7773:
7744:
7742:
7734:
7726:
7721:
7719:
7711:
7703:
7672:
7670:
7662:
7654:
7649:
7647:
7639:
7631:
7618:
7603:
7586:
7584:
7583:
7578:
7573:
7571:
7563:
7555:
7553:
7551:
7540:
7535:
7533:
7525:
7517:
7508:
7500:
7498:
7497:
7492:
7487:
7485:
7477:
7469:
7467:
7459:
7451:
7449:
7441:
7433:
7427:
7425:
7417:
7409:
7407:
7399:
7394:
7392:
7384:
7376:
7367:
7365:
7364:
7359:
7357:
7356:
7320:
7318:
7317:
7312:
7282:
7252:
7250:
7249:
7244:
7242:
7240:
7232:
7224:
7216:
7214:
7206:
7198:
7192:
7190:
7182:
7174:
7169:
7167:
7159:
7151:
7081:
7079:
7078:
7073:
7071:
7070:
7063:
7060:
7052:
7044:
7039:
7036:
7028:
7020:
7013:
7010:
7002:
6994:
6989:
6986:
6978:
6970:
6918:
6916:
6915:
6910:
6905:
6904:
6796:. As above, if
6795:
6793:
6792:
6787:
6776:
6775:
6755:
6753:
6752:
6747:
6742:
6741:
6666:
6664:
6663:
6658:
6570:
6568:
6567:
6562:
6554:
6552:
6544:
6527:
6522:
6520:
6512:
6504:
6479:
6477:
6476:
6471:
6463:
6461:
6453:
6445:
6440:
6438:
6430:
6413:
6396:
6394:
6393:
6388:
6386:
6385:
6377:
6361:
6359:
6358:
6353:
6351:
6350:
6319:
6318:
6310:
6273:
6271:
6270:
6265:
6263:
6260:
6245:
6243:
6242:
6237:
6204:
6202:
6201:
6196:
6194:
6191:
6176:
6174:
6173:
6168:
6166:
6163:
6138:
6136:
6135:
6130:
6128:
6125:
6106:
6104:
6103:
6098:
6096:
6093:
6066:
6064:
6063:
6058:
6050:
6048:
6040:
6032:
6027:
6025:
6017:
6009:
6004:
6002:
5994:
5986:
5981:
5979:
5971:
5963:
5958:
5956:
5948:
5940:
5935:
5933:
5925:
5917:
5912:
5910:
5902:
5894:
5889:
5887:
5879:
5871:
5844:
5842:
5841:
5836:
5784:
5782:
5781:
5776:
5768:
5766:
5765:
5764:
5751:
5747:
5746:
5736:
5731:
5729:
5728:
5727:
5714:
5710:
5709:
5699:
5668:
5666:
5665:
5660:
5655:
5650:
5648:
5640:
5632:
5627:
5622:
5620:
5612:
5604:
5533:stationary point
5498:
5496:
5495:
5490:
5476:
5474:
5473:
5468:
5456:real variables (
5450:
5448:
5447:
5442:
5425:
5423:
5422:
5417:
5415:
5414:
5406:
5395:
5393:
5392:
5387:
5373:
5371:
5370:
5365:
5352:
5350:
5349:
5344:
5339:
5337:
5329:
5321:
5316:
5314:
5306:
5298:
5290:in that case is
5288:
5286:
5285:
5280:
5267:
5265:
5264:
5259:
5248:
5246:
5245:
5244:
5236:
5229:
5221:
5212:
5210:
5209:
5204:
5199:
5195:
5194:
5192:
5181:
5173:
5171:
5160:
5153:
5145:
5140:
5138:
5137:
5136:
5128:
5118:
5110:
5106:
5105:
5103:
5092:
5084:
5082:
5071:
5064:
5056:
5051:
5049:
5038:
5025:
5023:
5022:
5017:
5004:
5002:
5001:
4996:
4983:
4981:
4980:
4975:
4958:
4953:
4939:
4931:, is defined by
4930:
4928:
4927:
4922:
4920:
4919:
4911:
4901:
4899:
4898:
4893:
4864:
4862:
4861:
4856:
4854:
4853:
4852:
4844:
4839:
4826:
4825:
4793:
4791:
4790:
4785:
4773:
4771:
4770:
4765:
4753:
4751:
4750:
4745:
4743:
4742:
4723:
4721:
4720:
4715:
4692:
4691:
4679:
4678:
4659:
4657:
4656:
4651:
4649:
4647:
4636:
4631:
4601:
4599:
4598:
4593:
4591:
4590:
4572:
4570:
4569:
4564:
4559:
4554:
4553:
4552:
4537:
4536:
4526:
4521:
4519:
4511:
4503:
4500:
4479:
4478:
4477:
4476:
4466:
4462:
4460:
4452:
4444:
4431:
4429:
4428:
4423:
4415:
4414:
4399:
4398:
4382:
4380:
4379:
4374:
4353:
4351:
4350:
4345:
4328:
4323:
4322:
4314:
4301:
4299:
4298:
4293:
4279:
4274:
4273:
4265:
4252:
4250:
4249:
4244:
4227:
4225:
4224:
4219:
4189:
4187:
4179:
4162:
4157:
4155:
4147:
4146:
4145:
4137:
4130:
4125:
4120:
4119:
4118:
4103:
4102:
4092:
4087:
4082:
4081:
4080:
4065:
4064:
4054:
4049:
4047:
4039:
4031:
4023:
4021:
4020:
4015:
3994:
3993:
3985:
3978:
3973:
3972:
3971:
3956:
3955:
3945:
3933:
3928:
3927:
3926:
3911:
3910:
3900:
3892:
3891:
3864:
3862:
3861:
3856:
3838:
3837:
3829:
3807:
3805:
3804:
3799:
3784:
3783:
3775:
3750:
3742:
3740:
3739:
3734:
3726:
3715:
3710:
3683:
3669:
3667:
3666:
3661:
3659:
3658:
3657:
3656:
3646:
3642:
3640:
3632:
3624:
3615:
3614:
3598:
3596:
3595:
3590:
3588:
3587:
3586:
3585:
3575:
3571:
3569:
3561:
3553:
3544:
3543:
3527:
3525:
3524:
3519:
3492:
3491:
3472:
3471:
3456:
3455:
3425:
3424:
3403:
3402:
3377:
3367:
3362:
3358:
3354:
3352:
3351:
3346:
3344:
3343:
3327:
3320:
3318:
3317:
3312:
3310:
3309:
3287:
3285:
3284:
3279:
3271:
3266:
3265:
3250:
3245:
3240:
3232:
3231:
3210:
3209:
3197:
3183:
3182:
3152:
3140:
3138:
3137:
3132:
3130:
3118:
3116:
3115:
3110:
3105:
3104:
3092:
3077:
3075:
3074:
3069:
3067:
3066:
3050:
3048:
3047:
3042:
3026:
3024:
3023:
3018:
3016:
3015:
2999:
2997:
2996:
2991:
2976:
2974:
2973:
2968:
2966:
2965:
2949:
2947:
2946:
2941:
2939:
2938:
2937:
2936:
2926:
2922:
2920:
2912:
2904:
2895:
2894:
2893:
2892:
2882:
2878:
2876:
2868:
2860:
2842:
2840:
2839:
2834:
2829:
2828:
2827:
2826:
2816:
2812:
2811:
2809:
2801:
2793:
2791:
2783:
2773:
2771:
2763:
2759:
2758:
2728:
2727:
2711:
2708:
2706:
2705:
2693:
2669:
2667:
2666:
2661:
2659:
2658:
2657:
2656:
2646:
2642:
2640:
2632:
2624:
2615:
2610:
2606:
2605:
2578:
2577:
2561:
2558:
2556:
2555:
2543:
2519:
2517:
2516:
2511:
2509:
2508:
2492:
2490:
2489:
2484:
2482:
2481:
2462:
2460:
2459:
2454:
2446:
2445:
2433:
2432:
2409:
2407:
2406:
2401:
2371:
2369:
2368:
2363:
2326:
2324:
2323:
2318:
2316:
2315:
2295:
2293:
2292:
2287:
2274:complex analysis
2268:
2266:
2265:
2260:
2258:
2257:
2241:
2239:
2238:
2233:
2231:
2226:
2222:
2221:
2194:
2193:
2177:
2174:
2172:
2171:
2159:
2136:
2135:
2123:
2108:
2106:
2105:
2100:
2098:
2097:
2082:
2081:
2069:
2068:
2052:
2050:
2049:
2044:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1988:
1986:
1985:
1980:
1914:
1912:
1911:
1906:
1894:
1892:
1891:
1886:
1874:
1872:
1871:
1866:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1791:
1789:
1788:
1783:
1710:
1699:
1668:
1666:
1665:
1660:
1652:
1629:
1605:
1603:
1602:
1597:
1589:
1569:
1545:
1543:
1542:
1537:
1532:
1531:
1475:
1466:
1464:
1463:
1458:
1453:
1451:
1443:
1435:
1430:
1429:
1427:
1419:
1411:
1397:
1370:complex analysis
1359:
1357:
1356:
1351:
1349:
1348:
1333:
1332:
1316:
1314:
1313:
1308:
1306:
1305:
1293:
1292:
1274:
1272:
1271:
1266:
1255:
1254:
1232:
1231:
1206:
1205:
1183:
1182:
1166:
1164:
1163:
1158:
1156:
1114:
1113:
1101:
1100:
1055:
1053:
1052:
1047:
1020:
1018:
1017:
1012:
985:
983:
982:
977:
960:
959:
947:
946:
934:
933:
887:
883:
881:
880:
875:
873:
872:
841:
839:
838:
833:
777:complex analysis
763:
750:
744:
734:
724:
718:
712:
706:
700:
686:
623:
617:
604:
589:
567:
558:
556:
555:
550:
545:
543:
535:
527:
519:
517:
509:
501:
491:
480:
471:
469:
468:
463:
461:
459:
451:
443:
438:
436:
428:
420:
410:
398:complex function
386:Bernhard Riemann
370:complex analysis
368:In the field of
358:
351:
344:
328:
327:
312:Karl Weierstrass
307:Bernhard Riemann
297:Jacques Hadamard
126:Imaginary number
106:
96:Complex analysis
90:
88:Complex analysis
79:
78:
21:
10968:
10967:
10963:
10962:
10961:
10959:
10958:
10957:
10923:
10922:
10921:
10916:
10896:
10807:Riemann surface
10782:Riemann problem
10698:
10693:
10637:
10632:
10626:
10586:
10566:
10564:Further reading
10561:
10555:
10507:10.2307/2321164
10488:
10483:
10482:
10471:
10467:
10459:
10455:
10447:
10443:
10429:
10425:
10421:, Theorem 11.2.
10417:
10413:
10402:Nomizu, Katsumi
10395:
10391:
10346:
10342:
10335:
10318:
10314:
10299:
10295:
10288:
10274:
10270:
10263:
10249:
10245:
10233:
10229:
10217:
10213:
10205:
10201:
10193:
10189:
10175:
10171:
10157:
10153:
10134:Euler, Leonhard
10131:
10127:
10102:
10098:
10093:
10071:
10062:
10056:
10036:
10029:
10007:
10006:
10002:
9997:
9994:
9993:
9963:
9959:
9955:
9921:
9920:
9916:
9911:
9908:
9907:
9886:
9881:
9880:
9866:
9863:
9862:
9855:
9834:
9833:
9822:
9814:
9811:
9794:
9786:
9783:
9780:
9779:
9768:
9760:
9757:
9743:
9735:
9732:
9725:
9724:
9722:
9719:
9718:
9697:
9696:
9681:
9677:
9665:
9661:
9658:
9657:
9642:
9638:
9626:
9622:
9615:
9614:
9593:
9589:
9577:
9573:
9564:
9560:
9545:
9541:
9529:
9525:
9516:
9512:
9501:
9498:
9497:
9478:
9474:
9472:
9469:
9468:
9451:
9447:
9445:
9442:
9441:
9425:
9424:
9409:
9405:
9397:
9393:
9386:
9374:
9370:
9364:
9360:
9354:
9350:
9349:
9347:
9334:
9330:
9324:
9320:
9308:
9304:
9296:
9292:
9288:
9276:
9272:
9266:
9262:
9256:
9252:
9251:
9249:
9236:
9232:
9226:
9222:
9213:
9212:
9200:
9196:
9190:
9186:
9168:
9164:
9158:
9154:
9145:
9141:
9135:
9131:
9112:
9105:
9103:
9100:
9099:
9072:
9069:
9068:
9037:
9034:
9033:
9032:. The function
9016:
9012:
9006:
9002:
8993:
8989:
8983:
8979:
8971:
8968:
8967:
8937:
8933:
8931:
8928:
8927:
8904:
8900:
8894:
8890:
8881:
8877:
8871:
8867:
8852:
8847:
8834:
8829:
8823:
8820:
8819:
8802:
8798:
8792:
8788:
8780:
8777:
8776:
8745:
8742:
8741:
8710:
8707:
8706:
8678:
8675:
8674:
8668:
8640:
8607:
8599:
8597:
8581:
8573:
8571:
8570:
8566:
8556:
8539:
8538:
8534:
8526:
8524:
8522:
8519:
8518:
8484:
8483:
8479:
8471:
8469:
8467:
8464:
8463:
8441:
8440:
8438:
8435:
8434:
8428:
8408:
8341:
8340:
8326:
8324:
8318:
8317:
8303:
8301:
8287:
8283:
8279:
8275:
8262:
8261:
8244:
8241:
8240:
8223:
8177:
8145:
8127:
8124:
8118:
8113:
8111:Generalizations
8076:
8064:
8063:
8050:
8048:
8031:
8030:
8023:
8019:
8010:
8006:
7993:
7988:
7969:
7968:
7961:
7957:
7952:
7949:
7948:
7866:
7865:
7836:
7835:
7831:
7823:
7821:
7819:
7816:
7815:
7799:
7787:
7771:
7770:
7745:
7735:
7727:
7725:
7712:
7704:
7702:
7699:
7698:
7673:
7663:
7655:
7653:
7640:
7632:
7630:
7626:
7624:
7621:
7620:
7605:
7590:
7564:
7556:
7554:
7544:
7539:
7526:
7518:
7516:
7514:
7511:
7510:
7504:
7478:
7470:
7468:
7458:
7442:
7434:
7432:
7418:
7410:
7408:
7398:
7385:
7377:
7375:
7373:
7370:
7369:
7349:
7345:
7334:
7331:
7330:
7288:
7285:
7284:
7256:
7233:
7225:
7223:
7207:
7199:
7197:
7183:
7175:
7173:
7160:
7152:
7150:
7148:
7145:
7144:
7117:
7065:
7064:
7053:
7045:
7042:
7040:
7029:
7021:
7018:
7015:
7014:
7003:
6995:
6992:
6990:
6979:
6971:
6968:
6961:
6960:
6934:
6931:
6930:
6923:Jacobian matrix
6899:
6898:
6877:
6876:
6851:
6850:
6827:
6824:
6823:
6771:
6767:
6765:
6762:
6761:
6736:
6735:
6730:
6724:
6723:
6715:
6705:
6704:
6696:
6693:
6692:
6685:
6632:
6629:
6628:
6603:magnetic fields
6575:Green's theorem
6545:
6528:
6526:
6513:
6505:
6503:
6501:
6498:
6497:
6454:
6446:
6444:
6431:
6414:
6412:
6410:
6407:
6406:
6376:
6375:
6373:
6370:
6369:
6368:) asserts that
6345:
6344:
6335:
6334:
6324:
6323:
6309:
6308:
6306:
6303:
6302:
6286:. Suppose that
6280:
6259:
6251:
6248:
6247:
6222:
6219:
6218:
6190:
6182:
6179:
6178:
6162:
6154:
6151:
6150:
6139:curves are the
6124:
6116:
6113:
6112:
6092:
6084:
6081:
6080:
6041:
6033:
6031:
6018:
6010:
6008:
5995:
5987:
5985:
5972:
5964:
5962:
5949:
5941:
5939:
5926:
5918:
5916:
5903:
5895:
5893:
5880:
5872:
5870:
5850:
5847:
5846:
5812:
5809:
5808:
5760:
5756:
5752:
5742:
5738:
5737:
5735:
5723:
5719:
5715:
5705:
5701:
5700:
5698:
5696:
5693:
5692:
5651:
5641:
5633:
5631:
5623:
5613:
5605:
5603:
5592:
5589:
5588:
5577:velocity vector
5557:stream function
5506:
5484:
5481:
5480:
5462:
5459:
5458:
5436:
5433:
5432:
5405:
5404:
5402:
5399:
5398:
5381:
5378:
5377:
5359:
5356:
5355:
5330:
5322:
5320:
5307:
5299:
5297:
5295:
5292:
5291:
5274:
5271:
5270:
5235:
5234:
5230:
5222:
5220:
5218:
5215:
5214:
5185:
5180:
5164:
5159:
5158:
5154:
5144:
5127:
5126:
5122:
5117:
5096:
5091:
5075:
5070:
5069:
5065:
5055:
5042:
5037:
5035:
5032:
5031:
5011:
5008:
5007:
4990:
4987:
4986:
4940:
4938:
4936:
4933:
4932:
4910:
4909:
4907:
4904:
4903:
4887:
4884:
4883:
4876:
4848:
4840:
4835:
4831:
4827:
4821:
4817:
4800:
4797:
4796:
4779:
4776:
4775:
4759:
4756:
4755:
4738:
4734:
4732:
4729:
4728:
4687:
4683:
4674:
4670:
4665:
4662:
4661:
4643:
4632:
4630:
4607:
4604:
4603:
4586:
4582:
4580:
4577:
4576:
4548:
4544:
4532:
4528:
4527:
4525:
4512:
4504:
4502:
4487:
4472:
4468:
4467:
4453:
4445:
4443:
4440:
4439:
4437:
4434:
4433:
4410:
4406:
4394:
4390:
4388:
4385:
4384:
4359:
4356:
4355:
4324:
4313:
4312:
4307:
4304:
4303:
4275:
4264:
4263:
4258:
4255:
4254:
4235:
4232:
4231:
4180:
4163:
4161:
4148:
4136:
4135:
4131:
4129:
4114:
4110:
4098:
4094:
4093:
4091:
4076:
4072:
4060:
4056:
4055:
4053:
4040:
4032:
4030:
4028:
4025:
4024:
3984:
3983:
3967:
3963:
3951:
3947:
3946:
3944:
3922:
3918:
3906:
3902:
3901:
3899:
3887:
3883:
3872:
3869:
3868:
3828:
3827:
3813:
3810:
3809:
3774:
3773:
3759:
3756:
3755:
3744:
3722:
3711:
3706:
3689:
3686:
3685:
3671:
3652:
3648:
3647:
3633:
3625:
3623:
3620:
3619:
3610:
3606:
3604:
3601:
3600:
3581:
3577:
3576:
3562:
3554:
3552:
3549:
3548:
3539:
3535:
3533:
3530:
3529:
3487:
3483:
3467:
3463:
3451:
3447:
3420:
3416:
3398:
3394:
3383:
3380:
3379:
3376:
3370:
3365:
3360:
3356:
3339:
3335:
3333:
3330:
3329:
3325:
3324:Conversely, if
3305:
3301:
3293:
3290:
3289:
3267:
3261:
3257:
3246:
3241:
3236:
3227:
3223:
3205:
3201:
3190:
3178:
3174:
3148:
3146:
3143:
3142:
3126:
3124:
3121:
3120:
3100:
3096:
3085:
3083:
3080:
3079:
3062:
3058:
3056:
3053:
3052:
3036:
3033:
3032:
3011:
3007:
3005:
3002:
3001:
2985:
2982:
2981:
2961:
2957:
2955:
2952:
2951:
2932:
2928:
2927:
2913:
2905:
2903:
2900:
2899:
2888:
2884:
2883:
2869:
2861:
2859:
2856:
2855:
2850:
2847:
2846:
2822:
2818:
2817:
2802:
2794:
2792:
2782:
2781:
2778:
2777:
2764:
2754:
2750:
2723:
2719:
2712:
2710:
2701:
2694:
2683:
2681:
2675:
2672:
2671:
2652:
2648:
2647:
2633:
2625:
2623:
2620:
2619:
2601:
2597:
2573:
2569:
2562:
2560:
2551:
2544:
2533:
2531:
2525:
2522:
2521:
2504:
2500:
2498:
2495:
2494:
2477:
2473:
2471:
2468:
2467:
2441:
2437:
2428:
2424:
2419:
2416:
2415:
2377:
2374:
2373:
2339:
2336:
2335:
2311:
2307:
2305:
2302:
2301:
2281:
2278:
2277:
2253:
2249:
2247:
2244:
2243:
2217:
2213:
2189:
2185:
2178:
2176:
2167:
2160:
2149:
2147:
2131:
2127:
2116:
2114:
2111:
2110:
2093:
2089:
2077:
2073:
2064:
2060:
2058:
2055:
2054:
2038:
2035:
2034:
2014:
2011:
2010:
1994:
1991:
1990:
1920:
1917:
1916:
1900:
1897:
1896:
1880:
1877:
1876:
1845:
1842:
1841:
1817:
1814:
1813:
1797:
1794:
1793:
1732:
1729:
1728:
1725:
1701:
1690:
1648:
1625:
1611:
1608:
1607:
1585:
1565:
1551:
1548:
1547:
1526:
1525:
1520:
1514:
1513:
1505:
1495:
1494:
1492:
1489:
1488:
1487:is of the form
1485:Jacobian matrix
1444:
1436:
1434:
1420:
1412:
1410:
1406:
1404:
1401:
1400:
1392:
1386:
1366:
1344:
1340:
1328:
1324:
1322:
1319:
1318:
1301:
1297:
1288:
1284:
1282:
1279:
1278:
1250:
1246:
1227:
1223:
1201:
1197:
1178:
1174:
1172:
1169:
1168:
1154:
1153:
1137:
1116:
1115:
1109:
1105:
1096:
1092:
1085:
1063:
1061:
1058:
1057:
1026:
1023:
1022:
991:
988:
987:
955:
951:
942:
938:
929:
925:
893:
890:
889:
885:
868:
864:
847:
844:
843:
812:
809:
808:
805:
785:
759:
746:
740:
730:
720:
714:
708:
702:
688:
637:
630:imaginary parts
619:
613:
591:
576:
536:
528:
526:
510:
502:
500:
498:
495:
494:
452:
444:
442:
429:
421:
419:
417:
414:
413:
388:, consist of a
382:Augustin Cauchy
362:
322:
232:Residue theorem
207:Local primitive
197:Zeros and poles
112:Complex numbers
82:
37:
28:
23:
22:
15:
12:
11:
5:
10966:
10956:
10955:
10950:
10945:
10940:
10935:
10918:
10917:
10915:
10914:
10901:
10898:
10897:
10895:
10894:
10889:
10884:
10879:
10874:
10869:
10864:
10859:
10854:
10849:
10844:
10839:
10834:
10829:
10824:
10819:
10814:
10809:
10804:
10799:
10797:Riemann sphere
10794:
10792:Riemann solver
10789:
10784:
10779:
10774:
10769:
10764:
10759:
10754:
10749:
10744:
10739:
10734:
10729:
10724:
10719:
10714:
10709:
10703:
10700:
10699:
10692:
10691:
10684:
10677:
10669:
10663:
10662:
10657:
10636:
10635:External links
10633:
10631:
10630:
10624:
10608:
10590:
10584:
10567:
10565:
10562:
10560:
10559:
10553:
10537:
10528:
10519:
10501:(4): 246–256.
10489:
10487:
10484:
10481:
10480:
10465:
10453:
10451:, p. 107.
10441:
10423:
10411:
10389:
10360:(5): 127–131.
10340:
10333:
10312:
10293:
10286:
10268:
10261:
10243:
10227:
10211:
10199:
10187:
10169:
10151:
10125:
10121:978-2012542839
10095:
10094:
10092:
10089:
10088:
10087:
10082:
10077:
10070:
10067:
10055:
10052:
10010:
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10001:
9975:
9970:
9966:
9962:
9958:
9954:
9951:
9948:
9945:
9942:
9939:
9936:
9933:
9930:
9924:
9919:
9915:
9889:
9884:
9879:
9876:
9873:
9870:
9854:
9851:
9837:
9828:
9825:
9820:
9817:
9810:
9807:
9800:
9797:
9792:
9789:
9782:
9781:
9774:
9771:
9766:
9763:
9756:
9749:
9746:
9741:
9738:
9731:
9730:
9728:
9700:
9695:
9692:
9689:
9684:
9680:
9676:
9673:
9668:
9664:
9660:
9659:
9656:
9653:
9650:
9645:
9641:
9637:
9634:
9629:
9625:
9621:
9620:
9618:
9613:
9610:
9607:
9604:
9601:
9596:
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9572:
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9481:
9477:
9454:
9450:
9423:
9420:
9417:
9412:
9408:
9400:
9396:
9392:
9389:
9383:
9377:
9373:
9367:
9363:
9357:
9353:
9345:
9342:
9337:
9333:
9327:
9323:
9319:
9316:
9311:
9307:
9299:
9295:
9291:
9285:
9279:
9275:
9269:
9265:
9259:
9255:
9247:
9244:
9239:
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9225:
9221:
9218:
9216:
9214:
9211:
9208:
9203:
9199:
9193:
9189:
9185:
9182:
9179:
9176:
9171:
9167:
9161:
9157:
9153:
9148:
9144:
9138:
9134:
9130:
9127:
9124:
9121:
9118:
9115:
9113:
9111:
9108:
9107:
9085:
9082:
9079:
9076:
9056:
9053:
9050:
9047:
9044:
9041:
9019:
9015:
9009:
9005:
9001:
8996:
8992:
8986:
8982:
8978:
8975:
8964:Dirac operator
8951:
8948:
8945:
8940:
8936:
8915:
8912:
8907:
8903:
8897:
8893:
8889:
8884:
8880:
8874:
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8866:
8863:
8860:
8855:
8850:
8846:
8842:
8837:
8832:
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8805:
8801:
8795:
8791:
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8764:
8761:
8758:
8755:
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8749:
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8723:
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8694:
8691:
8688:
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8624:
8620:
8613:
8610:
8605:
8602:
8596:
8593:
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8579:
8576:
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8560:
8555:
8546:
8543:
8537:
8532:
8529:
8506:
8503:
8500:
8491:
8488:
8482:
8477:
8474:
8448:
8445:
8432:d-bar operator
8427:
8424:
8407:
8404:
8396:
8395:
8344:
8339:
8336:
8333:
8325:
8323:
8320:
8319:
8316:
8313:
8310:
8302:
8299:
8293:
8290:
8286:
8282:
8278:
8274:
8271:
8268:
8267:
8265:
8260:
8257:
8254:
8251:
8248:
8144:as a function
8142:differentiable
8117:
8114:
8112:
8109:
8085:
8082:
8079:
8071:
8068:
8062:
8059:
8056:
8053:
8045:
8038:
8035:
8029:
8026:
8022:
8018:
8013:
8009:
8002:
7999:
7996:
7992:
7987:
7983:
7976:
7973:
7967:
7964:
7960:
7956:
7879:
7873:
7870:
7864:
7861:
7858:
7855:
7852:
7843:
7840:
7834:
7829:
7826:
7769:
7766:
7763:
7760:
7757:
7754:
7751:
7748:
7746:
7741:
7738:
7733:
7730:
7724:
7718:
7715:
7710:
7707:
7701:
7700:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7676:
7674:
7669:
7666:
7661:
7658:
7652:
7646:
7643:
7638:
7635:
7629:
7628:
7576:
7570:
7567:
7562:
7559:
7550:
7547:
7543:
7538:
7532:
7529:
7524:
7521:
7490:
7484:
7481:
7476:
7473:
7465:
7462:
7457:
7454:
7448:
7445:
7440:
7437:
7430:
7424:
7421:
7416:
7413:
7405:
7402:
7397:
7391:
7388:
7383:
7380:
7355:
7352:
7348:
7344:
7341:
7338:
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7239:
7236:
7231:
7228:
7222:
7219:
7213:
7210:
7205:
7202:
7195:
7189:
7186:
7181:
7178:
7172:
7166:
7163:
7158:
7155:
7116:
7113:
7096:commutes with
7069:
7059:
7056:
7051:
7048:
7041:
7035:
7032:
7027:
7024:
7017:
7016:
7009:
7006:
7001:
6998:
6991:
6985:
6982:
6977:
6974:
6967:
6966:
6964:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6908:
6903:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6878:
6875:
6872:
6869:
6866:
6863:
6860:
6857:
6856:
6854:
6849:
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6840:
6837:
6834:
6831:
6785:
6782:
6779:
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6740:
6734:
6731:
6729:
6726:
6725:
6722:
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6716:
6714:
6711:
6710:
6708:
6703:
6700:
6684:
6681:
6656:
6653:
6649:
6646:
6643:
6640:
6636:
6607:electrostatics
6599:magnetostatics
6595:potential flow
6591:fluid dynamics
6560:
6557:
6551:
6548:
6543:
6540:
6537:
6534:
6531:
6525:
6519:
6516:
6511:
6508:
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6443:
6437:
6434:
6429:
6426:
6423:
6420:
6417:
6383:
6380:
6349:
6343:
6340:
6337:
6336:
6333:
6330:
6329:
6327:
6322:
6316:
6313:
6279:
6276:
6258:
6255:
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6232:
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6123:
6120:
6091:
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6056:
6053:
6047:
6044:
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6016:
6013:
6007:
6001:
5998:
5993:
5990:
5984:
5978:
5975:
5970:
5967:
5961:
5955:
5952:
5947:
5944:
5938:
5932:
5929:
5924:
5921:
5915:
5909:
5906:
5901:
5898:
5892:
5886:
5883:
5878:
5875:
5869:
5866:
5863:
5860:
5857:
5854:
5834:
5831:
5828:
5825:
5822:
5819:
5816:
5774:
5771:
5763:
5759:
5755:
5750:
5745:
5741:
5734:
5726:
5722:
5718:
5713:
5708:
5704:
5658:
5654:
5647:
5644:
5639:
5636:
5630:
5626:
5619:
5616:
5611:
5608:
5602:
5599:
5596:
5505:
5502:
5492:{\textstyle y}
5488:
5470:{\textstyle x}
5466:
5444:{\textstyle z}
5440:
5412:
5409:
5389:{\textstyle z}
5385:
5367:{\textstyle f}
5363:
5342:
5336:
5333:
5328:
5325:
5319:
5313:
5310:
5305:
5302:
5282:{\textstyle f}
5278:
5257:
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5251:
5242:
5239:
5233:
5228:
5225:
5202:
5198:
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5179:
5176:
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5102:
5099:
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5015:
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4891:
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4847:
4843:
4838:
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4816:
4813:
4810:
4807:
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4783:
4763:
4741:
4737:
4713:
4710:
4707:
4704:
4701:
4698:
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4686:
4682:
4677:
4673:
4669:
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4642:
4639:
4635:
4629:
4626:
4623:
4620:
4617:
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4611:
4589:
4585:
4562:
4557:
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4535:
4531:
4524:
4518:
4515:
4510:
4507:
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4493:
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4486:
4482:
4475:
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4465:
4459:
4456:
4451:
4448:
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4421:
4418:
4413:
4409:
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4402:
4397:
4393:
4372:
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4366:
4363:
4343:
4340:
4337:
4334:
4331:
4327:
4320:
4317:
4311:
4291:
4288:
4285:
4282:
4278:
4271:
4268:
4262:
4242:
4239:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4192:
4186:
4183:
4178:
4175:
4172:
4169:
4166:
4160:
4154:
4151:
4143:
4140:
4134:
4128:
4123:
4117:
4113:
4109:
4106:
4101:
4097:
4090:
4085:
4079:
4075:
4071:
4068:
4063:
4059:
4052:
4046:
4043:
4038:
4035:
4012:
4009:
4006:
4003:
4000:
3997:
3991:
3988:
3982:
3976:
3970:
3966:
3962:
3959:
3954:
3950:
3943:
3940:
3937:
3931:
3925:
3921:
3917:
3914:
3909:
3905:
3898:
3895:
3890:
3886:
3882:
3879:
3876:
3854:
3851:
3847:
3844:
3841:
3835:
3832:
3826:
3823:
3820:
3817:
3797:
3794:
3790:
3787:
3781:
3778:
3772:
3769:
3766:
3763:
3732:
3729:
3725:
3721:
3718:
3714:
3709:
3705:
3702:
3699:
3696:
3693:
3655:
3651:
3645:
3639:
3636:
3631:
3628:
3622:
3618:
3613:
3609:
3584:
3580:
3574:
3568:
3565:
3560:
3557:
3551:
3547:
3542:
3538:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3490:
3486:
3482:
3479:
3476:
3470:
3466:
3462:
3459:
3454:
3450:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3423:
3419:
3415:
3412:
3409:
3406:
3401:
3397:
3393:
3390:
3387:
3374:
3342:
3338:
3308:
3304:
3300:
3297:
3277:
3274:
3270:
3264:
3260:
3256:
3253:
3249:
3244:
3239:
3235:
3230:
3226:
3222:
3219:
3216:
3213:
3208:
3204:
3200:
3196:
3193:
3189:
3186:
3181:
3177:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3151:
3129:
3108:
3103:
3099:
3095:
3091:
3088:
3065:
3061:
3040:
3014:
3010:
2989:
2980:(Note that if
2964:
2960:
2935:
2931:
2925:
2919:
2916:
2911:
2908:
2902:
2898:
2891:
2887:
2881:
2875:
2872:
2867:
2864:
2858:
2854:
2832:
2825:
2821:
2815:
2808:
2805:
2800:
2797:
2789:
2786:
2780:
2776:
2770:
2767:
2762:
2757:
2753:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2726:
2722:
2718:
2715:
2704:
2700:
2697:
2692:
2689:
2686:
2680:
2655:
2651:
2645:
2639:
2636:
2631:
2628:
2622:
2618:
2613:
2609:
2604:
2600:
2596:
2593:
2590:
2587:
2584:
2581:
2576:
2572:
2568:
2565:
2554:
2550:
2547:
2542:
2539:
2536:
2530:
2507:
2503:
2480:
2476:
2452:
2449:
2444:
2440:
2436:
2431:
2427:
2423:
2412:differentiable
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2333:real functions
2331:the bivariate
2329:if and only if
2314:
2310:
2285:
2256:
2252:
2229:
2225:
2220:
2216:
2212:
2209:
2206:
2203:
2200:
2197:
2192:
2188:
2184:
2181:
2170:
2166:
2163:
2158:
2155:
2152:
2146:
2142:
2139:
2134:
2130:
2126:
2122:
2119:
2109:is defined by
2096:
2092:
2088:
2085:
2080:
2076:
2072:
2067:
2063:
2046:{\textstyle f}
2042:
2022:{\textstyle y}
2018:
2002:{\textstyle x}
1998:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1908:{\textstyle y}
1904:
1888:{\textstyle x}
1884:
1864:
1861:
1858:
1855:
1852:
1849:
1821:
1805:{\textstyle u}
1801:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1724:
1721:
1658:
1655:
1651:
1647:
1644:
1641:
1638:
1635:
1632:
1628:
1624:
1621:
1618:
1615:
1595:
1592:
1588:
1584:
1581:
1578:
1575:
1572:
1568:
1564:
1561:
1558:
1555:
1535:
1530:
1524:
1521:
1519:
1516:
1515:
1512:
1509:
1506:
1504:
1501:
1500:
1498:
1479:
1478:
1469:
1467:
1456:
1450:
1447:
1442:
1439:
1433:
1426:
1423:
1418:
1415:
1409:
1385:
1382:
1365:
1362:
1347:
1343:
1339:
1336:
1331:
1327:
1304:
1300:
1296:
1291:
1287:
1264:
1261:
1258:
1253:
1249:
1244:
1241:
1238:
1235:
1230:
1226:
1221:
1218:
1215:
1212:
1209:
1204:
1200:
1195:
1192:
1189:
1186:
1181:
1177:
1152:
1149:
1146:
1143:
1140:
1138:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1117:
1112:
1108:
1104:
1099:
1095:
1091:
1088:
1086:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1065:
1045:
1042:
1039:
1036:
1033:
1030:
1010:
1007:
1004:
1001:
998:
995:
986:The real part
975:
972:
969:
966:
963:
958:
954:
950:
945:
941:
937:
932:
928:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
871:
867:
863:
860:
857:
854:
851:
831:
828:
825:
822:
819:
816:
804:
803:Simple example
801:
793:Leonhard Euler
784:
781:
607:differentiable
571:
570:
561:
559:
548:
542:
539:
534:
531:
525:
522:
516:
513:
508:
505:
484:
483:
474:
472:
458:
455:
450:
447:
441:
435:
432:
427:
424:
380:, named after
364:
363:
361:
360:
353:
346:
338:
335:
334:
333:
332:
317:
316:
315:
314:
309:
304:
299:
294:
289:
287:Leonhard Euler
284:
276:
275:
269:
268:
262:
261:
260:
259:
254:
249:
244:
239:
234:
229:
224:
222:Laurent series
219:
217:Winding number
214:
209:
204:
199:
191:
190:
184:
183:
182:
181:
176:
171:
166:
161:
153:
152:
146:
145:
144:
143:
138:
133:
128:
123:
115:
114:
108:
107:
99:
98:
92:
91:
26:
9:
6:
4:
3:
2:
10965:
10954:
10951:
10949:
10946:
10944:
10941:
10939:
10936:
10934:
10931:
10930:
10928:
10913:
10912:
10903:
10902:
10899:
10893:
10890:
10888:
10885:
10883:
10880:
10878:
10875:
10873:
10870:
10868:
10865:
10863:
10860:
10858:
10855:
10853:
10850:
10848:
10845:
10843:
10840:
10838:
10835:
10833:
10830:
10828:
10825:
10823:
10820:
10818:
10815:
10813:
10810:
10808:
10805:
10803:
10800:
10798:
10795:
10793:
10790:
10788:
10785:
10783:
10780:
10778:
10775:
10773:
10770:
10768:
10765:
10763:
10760:
10758:
10755:
10753:
10750:
10748:
10745:
10743:
10740:
10738:
10735:
10733:
10730:
10728:
10725:
10723:
10720:
10718:
10715:
10713:
10710:
10708:
10705:
10704:
10701:
10697:
10690:
10685:
10683:
10678:
10676:
10671:
10670:
10667:
10661:
10658:
10653:
10652:
10647:
10644:
10639:
10638:
10627:
10625:0-521-28763-4
10621:
10617:
10613:
10609:
10606:
10602:
10601:
10596:
10591:
10587:
10585:0-07-000657-1
10581:
10577:
10573:
10572:Ahlfors, Lars
10569:
10568:
10556:
10554:0-07-054234-1
10550:
10546:
10542:
10541:Rudin, Walter
10538:
10534:
10529:
10525:
10520:
10516:
10512:
10508:
10504:
10500:
10496:
10491:
10490:
10476:
10469:
10462:
10457:
10450:
10445:
10437:
10433:
10427:
10420:
10415:
10407:
10403:
10399:
10393:
10385:
10381:
10377:
10373:
10368:
10363:
10359:
10355:
10351:
10344:
10336:
10334:3-540-63640-4
10330:
10326:
10322:
10321:Pólya, George
10316:
10308:
10304:
10297:
10289:
10283:
10279:
10272:
10264:
10258:
10254:
10247:
10238:
10231:
10222:
10215:
10208:
10203:
10196:
10191:
10183:
10179:
10173:
10165:
10161:
10155:
10147:
10143:
10139:
10135:
10129:
10122:
10118:
10112:
10111:
10106:
10100:
10096:
10086:
10083:
10081:
10078:
10076:
10073:
10072:
10066:
10061:
10051:
10049:
10045:
10039:
10032:
10027:
10003:
9999:
9991:
9986:
9973:
9968:
9964:
9960:
9949:
9946:
9934:
9931:
9928:
9917:
9913:
9905:
9887:
9871:
9868:
9860:
9850:
9826:
9818:
9808:
9805:
9798:
9790:
9772:
9764:
9754:
9747:
9739:
9726:
9716:
9713:
9693:
9690:
9687:
9682:
9674:
9671:
9666:
9654:
9651:
9648:
9643:
9635:
9632:
9627:
9616:
9608:
9605:
9599:
9594:
9586:
9583:
9578:
9565:
9561:
9557:
9551:
9546:
9538:
9535:
9530:
9517:
9513:
9509:
9506:
9495:
9479:
9475:
9452:
9448:
9438:
9421:
9418:
9415:
9410:
9398:
9394:
9390:
9387:
9381:
9375:
9371:
9365:
9361:
9355:
9351:
9343:
9340:
9335:
9325:
9321:
9317:
9314:
9309:
9297:
9293:
9289:
9283:
9277:
9273:
9267:
9263:
9257:
9253:
9245:
9242:
9237:
9227:
9223:
9219:
9217:
9206:
9201:
9197:
9191:
9187:
9183:
9180:
9169:
9159:
9155:
9151:
9146:
9136:
9132:
9125:
9122:
9116:
9114:
9109:
9097:
9083:
9080:
9077:
9054:
9051:
9048:
9045:
9042:
9039:
9017:
9007:
9003:
8999:
8994:
8984:
8980:
8976:
8965:
8949:
8946:
8943:
8938:
8934:
8913:
8910:
8905:
8901:
8895:
8891:
8887:
8882:
8878:
8872:
8868:
8864:
8861:
8858:
8853:
8848:
8844:
8840:
8835:
8830:
8826:
8803:
8799:
8793:
8789:
8785:
8782:
8762:
8759:
8756:
8753:
8750:
8747:
8727:
8724:
8721:
8718:
8715:
8712:
8689:
8683:
8680:
8673:
8663:
8661:
8657:
8653:
8649:
8645:
8635:
8622:
8618:
8611:
8603:
8594:
8591:
8585:
8577:
8567:
8561:
8558:
8553:
8541:
8530:
8504:
8501:
8498:
8486:
8475:
8433:
8423:
8421:
8417:
8413:
8403:
8401:
8393:
8389:
8385:
8381:
8377:
8373:
8372:
8371:
8369:
8364:
8362:
8357:
8337:
8334:
8331:
8321:
8314:
8311:
8308:
8297:
8291:
8288:
8284:
8280:
8276:
8272:
8269:
8263:
8258:
8252:
8246:
8238:
8234:
8230:
8226:
8220:
8215:
8213:
8209:
8205:
8201:
8197:
8193:
8188:
8184:
8180:
8174:
8172:
8168:
8165:asserts that
8164:
8162:
8156:
8152:
8148:
8143:
8138:
8134:
8130:
8126:Suppose that
8123:
8108:
8106:
8102:
8083:
8080:
8077:
8066:
8060:
8057:
8054:
8051:
8043:
8033:
8027:
8024:
8020:
8016:
8011:
8007:
8000:
7997:
7994:
7990:
7985:
7981:
7971:
7965:
7962:
7958:
7954:
7946:
7942:
7938:
7934:
7930:
7926:
7925:
7920:
7915:
7913:
7909:
7905:
7901:
7897:
7893:
7868:
7862:
7859:
7853:
7850:
7838:
7827:
7813:
7807:
7803:
7795:
7791:
7784:
7764:
7761:
7758:
7752:
7749:
7747:
7739:
7731:
7722:
7716:
7708:
7692:
7689:
7686:
7680:
7677:
7675:
7667:
7659:
7650:
7644:
7636:
7616:
7612:
7608:
7601:
7597:
7593:
7587:
7574:
7568:
7560:
7548:
7545:
7541:
7536:
7530:
7522:
7507:
7501:
7488:
7482:
7474:
7463:
7460:
7455:
7452:
7446:
7438:
7428:
7422:
7414:
7403:
7400:
7395:
7389:
7381:
7353:
7350:
7346:
7342:
7339:
7336:
7328:
7324:
7305:
7299:
7296:
7280:
7276:
7272:
7268:
7264:
7260:
7253:
7237:
7229:
7220:
7217:
7211:
7203:
7193:
7187:
7179:
7170:
7164:
7156:
7143:, then so do
7142:
7138:
7134:
7133:
7128:
7127:
7122:
7112:
7110:
7106:
7101:
7099:
7095:
7091:
7087:
7082:
7067:
7057:
7049:
7033:
7025:
7007:
6999:
6983:
6975:
6962:
6957:
6951:
6948:
6945:
6939:
6936:
6928:
6924:
6919:
6906:
6901:
6892:
6889:
6886:
6880:
6870:
6867:
6864:
6858:
6852:
6847:
6841:
6838:
6835:
6829:
6821:
6819:
6815:
6811:
6807:
6803:
6799:
6783:
6780:
6777:
6772:
6768:
6759:
6743:
6738:
6732:
6727:
6720:
6717:
6712:
6706:
6701:
6698:
6690:
6680:
6678:
6674:
6670:
6654:
6651:
6647:
6644:
6641:
6638:
6634:
6627:
6623:
6619:
6615:
6610:
6608:
6604:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6571:
6558:
6555:
6549:
6538:
6535:
6523:
6517:
6509:
6495:
6491:
6487:
6486:
6480:
6467:
6464:
6458:
6450:
6441:
6435:
6424:
6421:
6404:
6400:
6378:
6367:
6366:
6347:
6341:
6338:
6331:
6325:
6320:
6311:
6301:
6297:
6293:
6289:
6285:
6275:
6256:
6253:
6233:
6230:
6227:
6216:
6212:
6208:
6187:
6184:
6159:
6156:
6149:
6144:
6143:of the flow.
6142:
6121:
6118:
6110:
6089:
6086:
6078:
6074:
6070:
6054:
6051:
6045:
6037:
6028:
6022:
6014:
6005:
5999:
5991:
5982:
5976:
5968:
5959:
5953:
5945:
5936:
5930:
5922:
5913:
5907:
5899:
5890:
5884:
5876:
5867:
5864:
5858:
5855:
5832:
5829:
5826:
5820:
5817:
5807:
5803:
5800:The function
5798:
5796:
5792:
5788:
5772:
5769:
5761:
5757:
5748:
5743:
5732:
5724:
5720:
5711:
5706:
5690:
5686:
5682:
5678:
5674:
5669:
5656:
5645:
5637:
5628:
5617:
5609:
5600:
5597:
5587:, defined by
5586:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5543:represents a
5542:
5534:
5530:
5526:
5522:
5518:
5514:
5510:
5501:
5499:
5486:
5477:
5464:
5455:
5451:
5438:
5429:
5407:
5396:
5383:
5374:
5361:
5340:
5334:
5326:
5317:
5311:
5308:
5303:
5300:
5289:
5276:
5255:
5252:
5249:
5237:
5226:
5200:
5196:
5189:
5177:
5174:
5168:
5155:
5149:
5146:
5141:
5129:
5111:
5107:
5100:
5088:
5085:
5079:
5066:
5060:
5057:
5052:
5046:
5029:
5013:
5005:
4992:
4971:
4968:
4965:
4962:
4959:
4950:
4947:
4944:
4941:
4912:
4889:
4881:
4871:
4869:
4845:
4836:
4832:
4828:
4822:
4818:
4814:
4808:
4802:
4781:
4761:
4739:
4735:
4725:
4708:
4705:
4702:
4696:
4688:
4684:
4680:
4675:
4671:
4640:
4637:
4627:
4621:
4618:
4615:
4609:
4587:
4583:
4573:
4560:
4555:
4549:
4545:
4541:
4538:
4533:
4529:
4522:
4516:
4508:
4497:
4491:
4480:
4473:
4469:
4463:
4457:
4454:
4449:
4446:
4419:
4416:
4411:
4407:
4403:
4400:
4395:
4391:
4370:
4364:
4341:
4338:
4335:
4332:
4325:
4315:
4289:
4286:
4283:
4276:
4266:
4240:
4228:
4215:
4209:
4206:
4203:
4190:
4184:
4173:
4164:
4158:
4152:
4138:
4126:
4121:
4115:
4111:
4107:
4104:
4099:
4095:
4088:
4083:
4077:
4073:
4069:
4066:
4061:
4057:
4050:
4044:
4036:
4007:
3998:
3995:
3986:
3974:
3968:
3964:
3960:
3957:
3952:
3948:
3941:
3938:
3929:
3923:
3919:
3915:
3912:
3907:
3903:
3896:
3888:
3884:
3877:
3866:
3852:
3845:
3842:
3839:
3830:
3821:
3818:
3795:
3788:
3785:
3776:
3767:
3764:
3752:
3748:
3730:
3719:
3707:
3700:
3691:
3682:
3678:
3674:
3653:
3649:
3643:
3637:
3629:
3616:
3611:
3607:
3582:
3578:
3572:
3566:
3558:
3545:
3540:
3536:
3512:
3503:
3500:
3497:
3488:
3484:
3480:
3477:
3468:
3464:
3460:
3452:
3448:
3441:
3438:
3432:
3426:
3421:
3417:
3410:
3407:
3399:
3395:
3388:
3373:
3368:
3340:
3336:
3322:
3306:
3302:
3295:
3275:
3262:
3258:
3254:
3251:
3242:
3228:
3224:
3220:
3217:
3206:
3202:
3194:
3191:
3187:
3179:
3175:
3168:
3165:
3159:
3153:
3101:
3097:
3089:
3086:
3063:
3059:
3038:
3030:
3012:
3008:
2987:
2978:
2962:
2958:
2933:
2929:
2923:
2917:
2909:
2896:
2889:
2885:
2879:
2873:
2865:
2852:
2843:
2830:
2823:
2819:
2813:
2806:
2798:
2787:
2784:
2774:
2768:
2765:
2755:
2751:
2744:
2741:
2735:
2732:
2729:
2724:
2720:
2713:
2698:
2695:
2690:
2684:
2653:
2649:
2643:
2637:
2629:
2616:
2611:
2602:
2598:
2591:
2588:
2582:
2579:
2574:
2570:
2563:
2548:
2545:
2540:
2534:
2505:
2501:
2478:
2474:
2464:
2450:
2442:
2438:
2434:
2429:
2425:
2413:
2394:
2391:
2388:
2385:
2379:
2356:
2353:
2350:
2347:
2341:
2334:
2330:
2312:
2308:
2299:
2283:
2275:
2270:
2254:
2250:
2227:
2218:
2214:
2207:
2204:
2198:
2195:
2190:
2186:
2179:
2164:
2161:
2156:
2150:
2140:
2132:
2128:
2120:
2117:
2094:
2090:
2086:
2083:
2078:
2074:
2070:
2065:
2061:
2040:
2032:
2016:
1996:
1973:
1970:
1967:
1961:
1958:
1952:
1949:
1946:
1943:
1937:
1934:
1928:
1922:
1902:
1882:
1862:
1859:
1856:
1853:
1850:
1847:
1839:
1835:
1819:
1799:
1776:
1770:
1767:
1764:
1761:
1755:
1749:
1746:
1740:
1734:
1720:
1716:
1714:
1708:
1704:
1697:
1693:
1688:
1684:
1680:
1676:
1672:
1656:
1649:
1645:
1639:
1636:
1633:
1626:
1622:
1616:
1613:
1593:
1586:
1582:
1576:
1573:
1566:
1562:
1556:
1553:
1533:
1528:
1522:
1517:
1510:
1507:
1502:
1496:
1486:
1477:
1470:
1468:
1454:
1448:
1440:
1431:
1424:
1416:
1407:
1399:
1398:
1395:
1391:
1390:Conformal map
1381:
1379:
1375:
1371:
1361:
1345:
1341:
1337:
1334:
1329:
1325:
1302:
1298:
1294:
1289:
1285:
1275:
1262:
1259:
1256:
1251:
1247:
1242:
1239:
1236:
1233:
1228:
1224:
1219:
1216:
1213:
1210:
1207:
1202:
1198:
1193:
1190:
1187:
1184:
1179:
1175:
1150:
1147:
1144:
1141:
1139:
1131:
1128:
1125:
1119:
1110:
1106:
1102:
1097:
1093:
1089:
1087:
1079:
1076:
1073:
1067:
1040:
1037:
1034:
1028:
1005:
1002:
999:
993:
973:
970:
967:
964:
961:
956:
952:
948:
943:
939:
935:
930:
922:
919:
916:
913:
907:
901:
895:
869:
865:
861:
855:
849:
829:
826:
823:
820:
817:
814:
807:Suppose that
800:
798:
794:
790:
780:
778:
773:
771:
767:
762:
757:
752:
749:
743:
738:
733:
728:
723:
717:
711:
705:
699:
695:
691:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
635:
631:
627:
622:
616:
610:
608:
602:
598:
594:
587:
583:
579:
569:
562:
560:
546:
540:
532:
523:
520:
514:
506:
493:
492:
489:
482:
475:
473:
456:
448:
439:
433:
425:
412:
411:
408:
405:
403:
399:
395:
391:
387:
383:
379:
375:
371:
359:
354:
352:
347:
345:
340:
339:
337:
336:
331:
326:
321:
320:
319:
318:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
283:
280:
279:
278:
277:
274:
271:
270:
267:
264:
263:
258:
255:
253:
250:
248:
247:Schwarz lemma
245:
243:
242:Conformal map
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
198:
195:
194:
193:
192:
189:
186:
185:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
156:
155:
154:
151:
148:
147:
142:
139:
137:
134:
132:
131:Complex plane
129:
127:
124:
122:
119:
118:
117:
116:
113:
110:
109:
105:
101:
100:
97:
94:
93:
89:
85:
81:
80:
74:
70:
66:
62:
58:
54:
50:
46:
41:
35:
30:
19:
10909:
10777:Riemann form
10706:
10649:
10615:
10612:Stewart, Ian
10598:
10575:
10544:
10532:
10523:
10498:
10494:
10474:
10468:
10463:, Theorem 9.
10456:
10444:
10435:
10426:
10414:
10405:
10392:
10357:
10353:
10343:
10327:. Springer.
10324:
10315:
10306:
10303:Klein, Felix
10296:
10277:
10271:
10252:
10246:
10236:
10230:
10220:
10214:
10202:
10190:
10181:
10172:
10163:
10154:
10145:
10141:
10128:
10109:
10099:
10063:
10037:
10030:
10025:
9989:
9987:
9858:
9856:
9717:
9714:
9496:
9440:Grouping by
9439:
9098:
8669:
8641:
8431:
8429:
8409:
8400:hypoelliptic
8397:
8387:
8383:
8379:
8375:
8365:
8360:
8358:
8239:
8232:
8228:
8224:
8218:
8216:
8207:
8203:
8199:
8195:
8186:
8182:
8178:
8175:
8170:
8166:
8159:
8154:
8150:
8146:
8136:
8132:
8128:
8125:
8104:
8100:
7940:
7932:
7928:
7923:
7918:
7916:
7911:
7907:
7903:
7899:
7895:
7891:
7811:
7805:
7801:
7793:
7789:
7785:
7614:
7610:
7606:
7599:
7595:
7591:
7588:
7505:
7502:
7278:
7274:
7270:
7266:
7262:
7258:
7254:
7140:
7136:
7130:
7124:
7118:
7102:
7097:
7093:
7089:
7085:
7083:
6926:
6920:
6822:
6817:
6813:
6809:
6805:
6801:
6797:
6757:
6686:
6621:
6617:
6611:
6583:conservative
6572:
6483:
6481:
6399:irrotational
6363:
6300:vector field
6295:
6291:
6287:
6281:
6210:
6206:
6148:level curves
6145:
6076:
6072:
6071:and that of
6068:
5801:
5799:
5786:
5684:
5676:
5672:
5670:
5584:
5572:
5568:
5564:
5563:) functions
5552:
5540:
5538:
5528:
5524:
5520:
5516:
5513:Contour plot
5479:
5457:
5453:
5431:
5427:
5376:
5354:
5269:
4985:
4877:
4726:
4574:
4229:
3867:
3753:
3746:
3680:
3676:
3672:
3371:
3364:
3323:
2979:
2844:
2465:
2271:
2030:
2029:. Then, the
1726:
1717:
1706:
1702:
1695:
1691:
1482:
1471:
1393:
1367:
1276:
806:
786:
774:
760:
753:
747:
741:
731:
721:
715:
709:
703:
697:
693:
689:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
638:
620:
614:
611:
600:
596:
592:
585:
581:
577:
574:
563:
487:
476:
406:
377:
367:
188:Basic theory
173:
87:
72:
68:
64:
60:
56:
52:
48:
44:
29:
10802:Riemann sum
10449:Looman 1923
10060:Pseudogroup
7931:, provided
7323:orthonormal
6616:. The pair
6109:streamlines
5806:dot product
5679:, with the
2053:at a point
1719:invariant.
1675:composition
612:Typically,
374:mathematics
302:Kiyoshi Oka
121:Real number
34:CR manifold
10927:Categories
10419:Rudin 1966
10223:. Chelsea.
10195:Rudin 1966
10091:References
10058:See also:
8642:Viewed as
8368:weak sense
8163:'s theorem
8120:See also:
6494:divergence
6490:solenoidal
6215:orthogonal
5795:divergence
5515:of a pair
4902:, denoted
10651:MathWorld
10605:EMS Press
10384:110258050
10376:0018-6368
9878:→
9875:Ω
9824:∂
9816:∂
9809:−
9796:∂
9788:∂
9770:∂
9762:∂
9745:∂
9737:∂
9679:∂
9663:∂
9640:∂
9636:−
9624:∂
9612:⇔
9591:∂
9575:∂
9562:σ
9543:∂
9539:−
9527:∂
9514:σ
9504:∇
9476:σ
9449:σ
9407:∂
9395:σ
9391:−
9382:⏟
9372:σ
9362:σ
9352:σ
9332:∂
9322:σ
9306:∂
9294:σ
9284:⏟
9274:σ
9264:σ
9254:σ
9234:∂
9224:σ
9198:σ
9188:σ
9166:∂
9156:σ
9143:∂
9133:σ
9120:∇
9075:∇
9014:∂
9004:σ
8991:∂
8981:σ
8977:≡
8974:∇
8947:−
8902:σ
8892:σ
8879:σ
8869:σ
8845:σ
8827:σ
8800:σ
8790:σ
8786:≡
8751:≡
8684:ℓ
8609:∂
8601:∂
8583:∂
8575:∂
8545:¯
8536:∂
8528:∂
8490:¯
8481:∂
8473:∂
8447:¯
8444:∂
8289:−
8281:−
8273:
8084:ζ
8081:−
8070:¯
8058:∧
8037:¯
8017:φ
8008:∬
7998:π
7975:¯
7972:ζ
7963:ζ
7872:¯
7854:φ
7842:¯
7833:∂
7825:∂
7753:β
7737:∂
7729:∂
7714:∂
7706:∂
7681:α
7665:∂
7657:∂
7651:−
7642:∂
7634:∂
7569:θ
7566:∂
7558:∂
7528:∂
7520:∂
7483:θ
7480:∂
7472:∂
7456:−
7444:∂
7436:∂
7423:θ
7420:∂
7412:∂
7387:∂
7379:∂
7354:θ
7303:∇
7294:∇
7235:∂
7227:∂
7221:−
7209:∂
7201:∂
7185:∂
7177:∂
7162:∂
7154:∂
7055:∂
7047:∂
7031:∂
7023:∂
7005:∂
6997:∂
6981:∂
6973:∂
6781:−
6718:−
6547:∂
6536:−
6530:∂
6515:∂
6507:∂
6456:∂
6448:∂
6442:−
6433:∂
6422:−
6416:∂
6382:¯
6339:−
6315:¯
6225:∇
6043:∂
6035:∂
6029:⋅
6020:∂
6012:∂
6006:−
5997:∂
5989:∂
5983:⋅
5974:∂
5966:∂
5951:∂
5943:∂
5937:⋅
5928:∂
5920:∂
5905:∂
5897:∂
5891:⋅
5882:∂
5874:∂
5862:∇
5859:⋅
5853:∇
5824:∇
5821:⋅
5815:∇
5785:That is,
5754:∂
5740:∂
5717:∂
5703:∂
5643:∂
5635:∂
5615:∂
5607:∂
5595:∇
5411:¯
5332:∂
5324:∂
5241:¯
5232:∂
5224:∂
5187:∂
5183:∂
5166:∂
5162:∂
5133:¯
5124:∂
5120:∂
5098:∂
5094:∂
5086:−
5077:∂
5073:∂
5044:∂
5040:∂
4966:−
4955:¯
4916:¯
4539:−
4514:Δ
4506:Δ
4495:→
4489:Δ
4368:→
4362:Δ
4339:−
4330:Δ
4319:¯
4310:Δ
4281:Δ
4270:¯
4261:Δ
4253:is real,
4238:Δ
4207:≠
4201:Δ
4182:Δ
4171:Δ
4165:η
4150:Δ
4142:¯
4133:Δ
4127:⋅
4067:−
4042:Δ
4034:Δ
4005:Δ
3999:η
3990:¯
3981:Δ
3936:Δ
3913:−
3875:Δ
3850:Δ
3834:¯
3825:Δ
3822:−
3816:Δ
3793:Δ
3780:¯
3771:Δ
3762:Δ
3728:→
3717:Δ
3698:Δ
3692:η
3635:∂
3627:∂
3564:∂
3556:∂
3510:Δ
3504:η
3495:Δ
3475:Δ
3439:−
3430:Δ
3386:Δ
3299:→
3273:→
3255:−
3221:−
3188:−
3166:−
2915:∂
2907:∂
2871:∂
2863:∂
2804:∂
2796:∂
2742:−
2699:∈
2688:→
2635:∂
2627:∂
2589:−
2549:∈
2538:→
2205:−
2165:∈
2154:→
1768:⋅
1713:conformal
1654:∂
1643:∂
1640:−
1631:∂
1620:∂
1591:∂
1580:∂
1571:∂
1560:∂
1508:−
1446:∂
1438:∂
1422:∂
1414:∂
1338:−
1211:−
1103:−
949:−
791:. Later,
725:are real
605:are real
538:∂
530:∂
524:−
512:∂
504:∂
454:∂
446:∂
431:∂
423:∂
10911:Category
10574:(1953).
10543:(1966).
10434:(1969).
10404:(1969).
10305:(1893).
10162:(1814).
10136:(1797).
10107:(1752).
10069:See also
9902:to be a
8656:solitons
8328:if
8312:≠
8305:if
8158:. Then
8149: :
8099:for all
6673:coclosed
6667:is both
6626:one-form
6577:and the
6496:-free):
5581:gradient
4230:Now, if
3195:′
3090:′
3029:Jacobian
2276:is that
2121:′
1679:rotation
10515:2321164
10486:Sources
10148:: 3–19.
8962:). The
8670:In the
8390:agrees
8161:Goursat
7937:closure
7129:) and (
6405:is 0):
5687:solves
5555:is its
1836:, be a
1683:scaling
1681:with a
783:History
634:complex
392:of two
71:, then
10622:
10582:
10551:
10513:
10382:
10374:
10331:
10284:
10259:
10241:, 2.14
10119:
10040:> 2
10024:, and
9988:where
8775:where
8517:where
7890:where
7509:gives
7123:. If (
6808:) and
6669:closed
6589:.) In
3754:Since
3684:, and
3528:where
1875:where
1792:where
1687:angles
1546:where
701:where
575:where
390:system
376:, the
273:People
10511:JSTOR
10380:S2CID
8926:, so
8235:/|z|)
7914:)/2.
6597:. In
6401:(its
6261:const
6192:const
6164:const
6126:const
6094:const
5789:is a
4868:below
1677:of a
632:of a
10620:ISBN
10580:ISBN
10549:ISBN
10372:ISSN
10329:ISBN
10300:See
10282:ISBN
10257:ISBN
10117:ISBN
9467:and
8658:and
8231:) =
8202:and
7902:and
7798:and
7604:and
7325:and
7139:and
6921:The
6671:and
6620:and
6492:(or
6403:curl
6290:and
6177:and
5675:and
5567:and
5519:and
5478:and
5006:and
4774:and
3808:and
3363:and
2410:are
2372:and
2009:and
1895:and
1832:are
1812:and
1727:Let
1606:and
1317:and
1056:are
768:and
745:and
719:and
707:and
673:) +
661:) =
649:) =
628:and
626:real
618:and
590:and
488:and
384:and
67:and
10503:doi
10362:doi
10033:= 2
9941:det
8818:, (
8374:If
8270:exp
8194:of
8185:+ i
8135:+ i
7939:of
7921:is
7917:If
7910:+ i
7906:= (
7898:+ i
7321:is
7269:),
6925:of
6679:).
6675:(a
6397:is
5583:of
5500:).
5454:two
5428:one
4882:of
4485:lim
3751:.
3749:→ 0
3743:as
3321:.)
3288:as
3051:at
3031:of
2679:lim
2529:lim
2414:at
2300:at
2296:is
2145:lim
2033:of
735:is
404:.
372:in
10929::
10648:.
10603:,
10597:,
10509:.
10499:85
10497:.
10400:;
10378:.
10370:.
10358:93
10356:.
10146:10
10144:.
10140:.
10050:.
9990:Df
9494::
8662:.
8214:.
8181:=
8153:→
8131:=
8107:.
8103:∈
7947:,
7933:𝜑
7919:𝜑
7904:𝜑
7894:=
7804:,
7800:β(
7792:,
7788:α(
7613:,
7598:,
7281:))
7277:,
7265:,
7132:1b
7126:1a
7111:.
7100:.
7094:Df
7088:,
6559:0.
6485:1a
6468:0.
6365:1b
5773:0.
5691::
5030:as
4960::=
3681:iy
3679:+
3675:=
3670:,
3599:,
2977:.
1715:.
1360:.
779:.
754:A
698:iy
696:+
692:=
681:,
675:iv
669:,
657:,
647:iy
645:+
599:,
584:,
566:1b
479:1a
86:→
10688:e
10681:t
10674:v
10654:.
10628:.
10588:.
10557:.
10517:.
10505::
10386:.
10364::
10337:.
10290:.
10265:.
10209:.
10197:.
10123:.
10038:n
10031:n
10026:I
10009:T
10004:f
10000:D
9974:I
9969:n
9965:/
9961:2
9957:)
9953:)
9950:f
9947:D
9944:(
9938:(
9935:=
9932:f
9929:D
9923:T
9918:f
9914:D
9888:n
9883:R
9872::
9869:f
9859:R
9827:x
9819:v
9806:=
9799:y
9791:u
9773:y
9765:v
9755:=
9748:x
9740:u
9727:{
9694:0
9691:=
9688:u
9683:y
9675:+
9672:v
9667:x
9655:0
9652:=
9649:v
9644:y
9633:u
9628:x
9617:{
9609:0
9606:=
9603:)
9600:u
9595:y
9587:+
9584:v
9579:x
9571:(
9566:2
9558:+
9555:)
9552:v
9547:y
9536:u
9531:x
9523:(
9518:1
9510:=
9507:f
9480:2
9453:1
9422:0
9419:=
9416:v
9411:y
9399:1
9388:=
9376:2
9366:1
9356:2
9344:+
9341:u
9336:y
9326:2
9318:+
9315:v
9310:x
9298:2
9290:=
9278:2
9268:1
9258:1
9246:+
9243:u
9238:x
9228:1
9220:=
9210:)
9207:v
9202:2
9192:1
9184:+
9181:u
9178:(
9175:)
9170:y
9160:2
9152:+
9147:x
9137:1
9129:(
9126:=
9123:f
9117:=
9110:0
9084:0
9081:=
9078:f
9055:v
9052:J
9049:+
9046:u
9043:=
9040:f
9018:y
9008:2
9000:+
8995:x
8985:1
8950:1
8944:=
8939:2
8935:J
8914:0
8911:=
8906:1
8896:2
8888:+
8883:2
8873:1
8865:,
8862:1
8859:=
8854:2
8849:2
8841:=
8836:2
8831:1
8804:2
8794:1
8783:J
8763:y
8760:J
8757:+
8754:x
8748:z
8728:y
8725:i
8722:+
8719:x
8716:=
8713:z
8693:)
8690:2
8687:(
8681:C
8623:.
8619:)
8612:y
8604:f
8595:i
8592:+
8586:x
8578:f
8568:(
8562:2
8559:1
8554:=
8542:z
8531:f
8505:,
8502:0
8499:=
8487:z
8476:f
8388:f
8384:C
8380:z
8378:(
8376:f
8361:z
8338:0
8335:=
8332:z
8322:0
8315:0
8309:z
8298:)
8292:4
8285:z
8277:(
8264:{
8259:=
8256:)
8253:z
8250:(
8247:f
8233:z
8229:z
8227:(
8225:f
8219:f
8208:f
8204:y
8200:x
8196:f
8187:v
8183:u
8179:f
8171:f
8167:f
8155:R
8151:R
8147:f
8137:v
8133:u
8129:f
8105:D
8101:ζ
8078:z
8067:z
8061:d
8055:z
8052:d
8044:)
8034:z
8028:,
8025:z
8021:(
8012:D
8001:i
7995:2
7991:1
7986:=
7982:)
7966:,
7959:(
7955:f
7941:D
7929:D
7924:C
7912:β
7908:α
7900:v
7896:u
7892:f
7878:)
7869:z
7863:,
7860:z
7857:(
7851:=
7839:z
7828:f
7812:R
7808:)
7806:y
7802:x
7796:)
7794:y
7790:x
7768:)
7765:y
7762:,
7759:x
7756:(
7750:=
7740:x
7732:v
7723:+
7717:y
7709:u
7696:)
7693:y
7690:,
7687:x
7684:(
7678:=
7668:y
7660:v
7645:x
7637:u
7617:)
7615:y
7611:x
7609:(
7607:v
7602:)
7600:y
7596:x
7594:(
7592:u
7575:.
7561:f
7549:r
7546:i
7542:1
7537:=
7531:r
7523:f
7506:f
7489:.
7475:u
7464:r
7461:1
7453:=
7447:r
7439:v
7429:,
7415:v
7404:r
7401:1
7396:=
7390:r
7382:u
7351:i
7347:e
7343:r
7340:=
7337:z
7309:)
7306:s
7300:,
7297:n
7291:(
7279:y
7275:x
7273:(
7271:s
7267:y
7263:x
7261:(
7259:n
7257:(
7238:s
7230:u
7218:=
7212:n
7204:v
7194:,
7188:s
7180:v
7171:=
7165:n
7157:u
7141:v
7137:u
7098:J
7090:v
7086:u
7068:]
7058:y
7050:v
7034:x
7026:v
7008:y
7000:u
6984:x
6976:u
6963:[
6958:=
6955:)
6952:y
6949:,
6946:x
6943:(
6940:f
6937:D
6927:f
6907:.
6902:]
6896:)
6893:y
6890:,
6887:x
6884:(
6881:v
6874:)
6871:y
6868:,
6865:x
6862:(
6859:u
6853:[
6848:=
6845:)
6842:y
6839:,
6836:x
6833:(
6830:f
6818:y
6816:,
6814:x
6812:(
6810:v
6806:y
6804:,
6802:x
6800:(
6798:u
6784:I
6778:=
6773:2
6769:J
6758:J
6744:.
6739:]
6733:0
6728:1
6721:1
6713:0
6707:[
6702:=
6699:J
6655:y
6652:d
6648:u
6645:+
6642:x
6639:d
6635:v
6622:v
6618:u
6556:=
6550:y
6542:)
6539:v
6533:(
6524:+
6518:x
6510:u
6465:=
6459:y
6451:u
6436:x
6428:)
6425:v
6419:(
6379:f
6348:]
6342:v
6332:u
6326:[
6321:=
6312:f
6296:R
6292:v
6288:u
6257:=
6254:u
6234:0
6231:=
6228:u
6211:v
6207:u
6188:=
6185:v
6160:=
6157:u
6122:=
6119:u
6090:=
6087:v
6077:u
6073:v
6069:u
6055:0
6052:=
6046:x
6038:v
6023:x
6015:u
6000:x
5992:v
5977:x
5969:u
5960:=
5954:y
5946:v
5931:y
5923:u
5914:+
5908:x
5900:v
5885:x
5877:u
5868:=
5865:v
5856:u
5845:(
5833:0
5830:=
5827:v
5818:u
5802:v
5787:u
5770:=
5762:2
5758:y
5749:u
5744:2
5733:+
5725:2
5721:x
5712:u
5707:2
5685:u
5677:v
5673:u
5657:.
5653:j
5646:y
5638:u
5629:+
5625:i
5618:x
5610:u
5601:=
5598:u
5585:u
5573:u
5569:v
5565:u
5553:v
5541:u
5529:u
5525:v
5521:v
5517:u
5487:y
5465:x
5439:z
5408:z
5384:z
5362:f
5341:.
5335:z
5327:f
5318:=
5312:z
5309:d
5304:f
5301:d
5277:f
5256:,
5253:0
5250:=
5238:z
5227:f
5201:,
5197:)
5190:y
5178:i
5175:+
5169:x
5156:(
5150:2
5147:1
5142:=
5130:z
5112:,
5108:)
5101:y
5089:i
5080:x
5067:(
5061:2
5058:1
5053:=
5047:z
5014:y
4993:x
4972:y
4969:i
4963:x
4951:y
4948:i
4945:+
4942:x
4913:z
4890:z
4850:|
4846:z
4842:|
4837:/
4833:i
4829:e
4823:2
4819:z
4815:=
4812:)
4809:z
4806:(
4803:f
4782:v
4762:u
4740:0
4736:z
4712:)
4709:0
4706:,
4703:0
4700:(
4697:=
4694:)
4689:0
4685:y
4681:,
4676:0
4672:x
4668:(
4645:|
4641:y
4638:x
4634:|
4628:=
4625:)
4622:y
4619:,
4616:x
4613:(
4610:f
4588:0
4584:z
4561:.
4556:2
4550:y
4546:f
4542:i
4534:x
4530:f
4523:=
4517:z
4509:f
4498:0
4492:z
4481:=
4474:0
4470:z
4464:|
4458:z
4455:d
4450:f
4447:d
4420:0
4417:=
4412:y
4408:f
4404:i
4401:+
4396:x
4392:f
4371:0
4365:z
4342:1
4336:=
4333:z
4326:/
4316:z
4290:1
4287:=
4284:z
4277:/
4267:z
4241:z
4216:.
4213:)
4210:0
4204:z
4198:(
4191:,
4185:z
4177:)
4174:z
4168:(
4159:+
4153:z
4139:z
4122:2
4116:y
4112:f
4108:i
4105:+
4100:x
4096:f
4089:+
4084:2
4078:y
4074:f
4070:i
4062:x
4058:f
4051:=
4045:z
4037:f
4011:)
4008:z
4002:(
3996:+
3987:z
3975:2
3969:y
3965:f
3961:i
3958:+
3953:x
3949:f
3942:+
3939:z
3930:2
3924:y
3920:f
3916:i
3908:x
3904:f
3897:=
3894:)
3889:0
3885:z
3881:(
3878:f
3853:y
3846:i
3843:2
3840:=
3831:z
3819:z
3796:x
3789:2
3786:=
3777:z
3768:+
3765:z
3747:z
3745:Δ
3731:0
3724:|
3720:z
3713:|
3708:/
3704:)
3701:z
3695:(
3677:x
3673:z
3654:0
3650:z
3644:|
3638:y
3630:f
3617:=
3612:y
3608:f
3583:0
3579:z
3573:|
3567:x
3559:f
3546:=
3541:x
3537:f
3516:)
3513:z
3507:(
3501:+
3498:y
3489:y
3485:f
3481:+
3478:x
3469:x
3465:f
3461:=
3458:)
3453:0
3449:z
3445:(
3442:f
3436:)
3433:z
3427:+
3422:0
3418:z
3414:(
3411:f
3408:=
3405:)
3400:0
3396:z
3392:(
3389:f
3375:0
3372:z
3366:y
3361:x
3357:f
3341:0
3337:z
3326:f
3307:0
3303:z
3296:z
3276:0
3269:|
3263:0
3259:z
3252:z
3248:|
3243:/
3238:|
3234:)
3229:0
3225:z
3218:z
3215:(
3212:)
3207:0
3203:z
3199:(
3192:f
3185:)
3180:0
3176:z
3172:(
3169:f
3163:)
3160:z
3157:(
3154:f
3150:|
3128:C
3107:)
3102:0
3098:z
3094:(
3087:f
3064:0
3060:z
3039:f
3013:0
3009:z
2988:f
2963:0
2959:z
2934:0
2930:z
2924:|
2918:y
2910:f
2897:=
2890:0
2886:z
2880:|
2874:x
2866:f
2853:i
2831:.
2824:0
2820:z
2814:|
2807:y
2799:f
2788:i
2785:1
2775:=
2769:h
2766:i
2761:)
2756:0
2752:z
2748:(
2745:f
2739:)
2736:h
2733:i
2730:+
2725:0
2721:z
2717:(
2714:f
2703:R
2696:h
2691:0
2685:h
2654:0
2650:z
2644:|
2638:x
2630:f
2617:=
2612:h
2608:)
2603:0
2599:z
2595:(
2592:f
2586:)
2583:h
2580:+
2575:0
2571:z
2567:(
2564:f
2553:R
2546:h
2541:0
2535:h
2506:0
2502:z
2479:0
2475:z
2451:,
2448:)
2443:0
2439:y
2435:,
2430:0
2426:x
2422:(
2398:)
2395:y
2392:i
2389:+
2386:x
2383:(
2380:v
2360:)
2357:y
2354:i
2351:+
2348:x
2345:(
2342:u
2313:0
2309:z
2284:f
2255:0
2251:z
2228:h
2224:)
2219:0
2215:z
2211:(
2208:f
2202:)
2199:h
2196:+
2191:0
2187:z
2183:(
2180:f
2169:C
2162:h
2157:0
2151:h
2141:=
2138:)
2133:0
2129:z
2125:(
2118:f
2095:0
2091:y
2087:i
2084:+
2079:0
2075:x
2071:=
2066:0
2062:z
2041:f
2017:y
1997:x
1977:)
1974:y
1971:,
1968:x
1965:(
1962:f
1959:=
1956:)
1953:y
1950:i
1947:+
1944:x
1941:(
1938:f
1935:=
1932:)
1929:z
1926:(
1923:f
1903:y
1883:x
1863:y
1860:i
1857:+
1854:x
1851:=
1848:z
1820:v
1800:u
1780:)
1777:z
1774:(
1771:v
1765:i
1762:+
1759:)
1756:z
1753:(
1750:u
1747:=
1744:)
1741:z
1738:(
1735:f
1709:)
1707:z
1705:(
1703:f
1698:)
1696:z
1694:(
1692:f
1657:y
1650:/
1646:u
1637:=
1634:x
1627:/
1623:v
1617:=
1614:b
1594:y
1587:/
1583:v
1577:=
1574:x
1567:/
1563:u
1557:=
1554:a
1534:,
1529:)
1523:a
1518:b
1511:b
1503:a
1497:(
1476:)
1474:2
1472:(
1455:.
1449:y
1441:f
1432:=
1425:x
1417:f
1408:i
1346:x
1342:v
1335:=
1330:y
1326:u
1303:y
1299:v
1295:=
1290:x
1286:u
1263:x
1260:2
1257:=
1252:y
1248:v
1243:;
1240:y
1237:2
1234:=
1229:x
1225:v
1220:;
1217:y
1214:2
1208:=
1203:y
1199:u
1194:;
1191:x
1188:2
1185:=
1180:x
1176:u
1151:y
1148:x
1145:2
1142:=
1135:)
1132:y
1129:,
1126:x
1123:(
1120:v
1111:2
1107:y
1098:2
1094:x
1090:=
1083:)
1080:y
1077:,
1074:x
1071:(
1068:u
1044:)
1041:y
1038:,
1035:x
1032:(
1029:v
1009:)
1006:y
1003:,
1000:x
997:(
994:u
974:y
971:x
968:i
965:2
962:+
957:2
953:y
944:2
940:x
936:=
931:2
927:)
923:y
920:i
917:+
914:x
911:(
908:=
905:)
902:z
899:(
896:f
886:z
870:2
866:z
862:=
859:)
856:z
853:(
850:f
830:y
827:i
824:+
821:x
818:=
815:z
761:C
748:v
742:u
732:f
722:v
716:u
710:y
704:x
694:x
690:z
685:)
683:y
679:x
677:(
671:y
667:x
665:(
663:u
659:y
655:x
653:(
651:f
643:x
641:(
639:f
621:v
615:u
603:)
601:y
597:x
595:(
593:v
588:)
586:y
582:x
580:(
578:u
568:)
564:(
547:,
541:x
533:v
521:=
515:y
507:u
481:)
477:(
457:y
449:v
440:=
434:x
426:u
357:e
350:t
343:v
73:f
69:z
65:X
61:z
57:f
53:f
49:z
45:X
36:.
20:)
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