View source for Cauchy-Riemann equations

{{MSC|30-XX|32-XX}}
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Also known as Cauchy-Riemann conditions and D'Alembert-Euler conditions, they are the partial differential equations that must be satisfied by the real and imaginary parts of a complex-valued function $f$ of one (or several) complex variable so that $f$ is [[Holomorphic function|holomorphic]]. 


===One complex variable===
More precisely, assume $D\subset \mathbb C$ is some open set and $f: D \to \mathbb C$ a map with real and imaginary parts given by $u$ and $v$ (i.e. $f(z) = u (z) + i v (z)$). If we introduce the real variables $x,y$ so that $z= x+iy$, we can consider $u$ and $v$ as real functions on a domain in $\mathbb R^2$. If $u$ and $v$ are differentiable (in the real-variable sense), then they solve the Cauchy-Riemann equations if 
\begin{equation}\label{e:CR}
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad
\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}\, .
\end{equation}
Such equations are equivalent to the fact that $f$ is complex-differentiable at every point $z_0\in D$, where its complex derivative is defined by
\[
f' (z_0) = \lim_{w\in \mathbb C, w\to 0} \frac{f(z_0+w)-f(z_0)}{w}\, .
\]
It then turns out that
\begin{equation}\label{e:f'}
f' = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}
\end{equation}
(and evidently we can use \eqref{e:CR} to derive three other similar formulas).

====Harmonicity and conjugacy====
Any pair of solutions of the Cauchy-Riemann equations turn out to be infinitely differentiable and in fact [[Analytic function|analytic]]. Moreover they are also [[Conjugate harmonic functions|conjugate]] [[Harmonic function|harmonic functions]]. Viceversa if $u$ (or $v$) is a given harmonic function on a [[Simply-connected domain|simply connected]] open $D$, then there is a conjugate harmonic function to $u$, that is a $v$ satisfying \eqref{e:CR}. Such $v$ is unique up to addition of a constant.

====Different systems of coordinates====
The conditions \eqref{e:CR} can be written, equivalently, for any two orthogonal directions $s$ and $n$, with the same mutual orientation as the $x$- and $y$-axes, in the form:
\[
\frac{\partial u}{\partial s} = \frac{\partial v}{\partial n} \qquad
\frac{\partial u}{\partial n} = - \frac{\partial v}{\partial s}\, .
\]
For example,  in polar coordinates $r, \phi$, for $r\neq 0$ the Cauchy-Riemann equations read:
\[
\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \phi} \qquad
\frac{1}{r} \frac{\partial u}{\partial \phi} = - \frac{\partial v}{\partial r}\, .
\]

====$\frac{\partial}{\partial \bar{z}}$ and $\frac{\partial}{\partial z}$ operators====
Defining the complex differential operators by
\[
\frac{\partial}{\partial \bar{z}}=\frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)\qquad\mbox{and}\qquad
\frac{\partial}{\partial z}=\frac{1}{2} \left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right)\, ,
\]
one can rewrite the Cauchy–Riemann equations \eqref{e:CR} as
\[
\frac{\partial f}{\partial \bar{z}} = 0
\]
and the identity \eqref{e:f'} as
\[
f' = \frac{\partial f}{\partial z}\, .
\]

====Conformality====
The Cauchy-Riemann equations are equivalent to the fact that the map $(u,v): D \to \mathbb R^2$ is [[Conformal mapping|conformal]], i.e. it preserves the angles and (locally where it is injective) the orientation. Such condition can indeed be expressed at the differential level with the property that at each point $(x_0, y_0)$ the [[Jacobian|Jacobian matrix]] of $(u,v)$ is a multiple of a rotation. In turn this can be easily seen to be equivalent to \eqref{e:CR}.

===Several complex variables===
For analytic functions of several complex variables $z = (z_1, \ldots, z_n)$, with $z_k = x_k + iy_k$ , the  Cauchy–Riemann equations is given by the following system of partial differential  equations (overdetermined when $n>1$) for the  functions
\begin{align*}
& u (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Re}\, f (x_1+iy_1, \ldots, x_n + iy_n)\\
& v (x_1, \ldots, x_n, y_1, \ldots , y_n) = {\rm Im}\, f (x_1+iy_1, \ldots, x_n + iy_n):
\end{align*}
\begin{equation}\label{e:CR-sys}
\frac{\partial u}{\partial x_k} = \frac{\partial v}{\partial y_k} \qquad
\frac{\partial u}{\partial y_k} = - \frac{\partial v}{\partial x_k}\,  \qquad k = 1, \ldots, n\, ,
\end{equation}
or, in terms of the complex differential operators:
\[
\frac{\partial f}{\partial \bar{z}_k} = 0\, .
\]
Each of the  two functions $u$ and $v$ satisfying the  conditions \eqref{e:CR-sys} (which as in the one-variable case, turn out to be infinitely differentiable and analytic) is a [[Pluriharmonic function|pluriharmonic function]] of  the variables $x_k$ and $y_k$. When $n>1$ the pluriharmonic  functions constitute a proper subclass of the class of harmonic  functions. The conditions \eqref{e:CR-sys} are conjugacy conditions for two  pluriharmonic functions $u$ and $v$: knowing one of  them, one can determine the other by integration (up to addition of a constant in each connected component of the domain of definition).

===Historical remarks===
The conditions \eqref{e:CR} apparently occurred for the first time in the works of [[DAlembert|J. d'Alembert]] {{Cite|DA}}. Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 {{Cite|Eu}}. A.L. Cauchy utilized the conditions \eqref{e:CR} to construct the theory of functions, beginning with a memoir presented to the Paris Academy in 1814 (see {{Cite|Ca}}). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates to 1851 (see {{Cite|Ri}}).

===References===
{|
|-
|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
|-
|valign="top"|{{Ref|Ca}}|| A.L. Cauchy,   "Mémoire sur les intégrales définies" , ''Oeuvres complètes Ser. 1'' , '''1''' , Paris  (1882)  pp. 319–506
|-
|valign="top"|{{Ref|DA}}|| J. d'Alembert,   "Essai d'une nouvelle théorie de la résistance des fluides" , Paris  (1752)
|-
|valign="top"|{{Ref|Eu}}|| L. Euler,   ''Nova Acta Acad. Sci. Petrop.'' , '''10'''  (1797)  pp. 3–19
|-
|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a  complex variable" ,  '''1–3''' , Chelsea (1977) (Translated from  Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
|-
|valign="top"|{{Ref|Ri}}|| B. Riemann, "Grundlagen für eine allgemeine Theorie der Funktionen einer  veränderlichen komplexen Grösse"  H. Weber (ed.) , ''Riemann's  gesammelte math. Werke'' , Dover, reprint  (1953)  pp. 3–48 
|-
|valign="top"|{{Ref|Sh}}|| B.V. Shabat, "Introduction of complex  analysis" , '''1–2''' , Moscow  (1976) (In Russian) {{MR|}}  {{ZBL|0799.32001}} {{ZBL|0732.32001}}  {{ZBL|0732.30001}}  {{ZBL|0578.32001}} {{ZBL|0574.30001}}  
|-
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