Cauchy-Riemann equations

2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]

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.

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. Moreover they are also conjugate harmonic functions. Viceversa if $u$ (or $v$) is a given harmonic function on a 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, 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 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 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 J. d'Alembert [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 [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 [Ca]). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates to 1851 (see [Ri]).

References

[Al]	L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Ca]	A.L. Cauchy, "Mémoire sur les intégrales définies" , Oeuvres complètes Ser. 1 , 1 , Paris (1882) pp. 319–506
[DA]	J. d'Alembert, "Essai d'une nouvelle théorie de la résistance des fluides" , Paris (1752)
[Eu]	L. Euler, Nova Acta Acad. Sci. Petrop. , 10 (1797) pp. 3–19
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[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
[Sh]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001