Cauchy-Riemann equations

In complex analysis, the Cauchy-Riemann equations are one of the of the basic objects of the theory: they are a system of $\scriptstyle 2n$ partial differential equations, where $\scriptstyle n$ is the dimension of the complex ambient space ℂⁿ considered. Precisely, their homogeneous form express a necessary and sufficient condition between the real and imaginary part of a complex valued function of $\scriptstyle 2n$ real variables for the given function to be a holomorphic one. They are named after Augustin-Louis Cauchy and Bernhard Riemann who were the first ones to study and use such equations as a mathematical object "per se", creating a new theory. These equations are sometimes referred as Cauchy-Riemann conditions or Cauchy-Riemann system: the partial differential operator appearing on the left side of these equations is usually called the Cauchy-Riemann operator.

Formal definition

In the following text, it is assumed that ℂⁿ≡ℝ²ⁿ, identifying the points of the euclidean spaces on the complex and real fields as follows

$z=(z_{1},\dots ,z_{n})\equiv (x_{1},y_{1},\dots ,x_{n},y_{n})$

The subscript is omitted when n=1.

The Cauchy-Riemann equations in ℂ (n=1)

Let f(x, y) = u(x, y) + iv(x, y) a complex valued differentiable function. Then f satisfies the homogeneous Cauchy-Riemann equations if and only if

$\left\{{\begin{array}{l}{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\\{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\\\end{array}}\right.$

Using Wirtinger derivatives these equation can be written in the following more compact form:

${\frac {\partial f}{\partial {\bar {z}}}}=0$

The Cauchy-Riemann equations in ℂⁿ (n>1)

Let f(x₁, y₁,...,x_n, y_n) = u(x₁, y₁,...,x_n, y_n) + iv(x₁, y₁,...,x_n, y_n) a complex valued differentiable function. Then f satisfies the homogeneous Cauchy-Riemann equations if and only if

$\left\{{\begin{array}{l}{\frac {\partial u}{\partial x_{1}}}={\frac {\partial v}{\partial y_{1}}}\\{\frac {\partial u}{\partial y_{1}}}=-{\frac {\partial v}{\partial x_{1}}}\\\qquad \vdots \\{\frac {\partial u}{\partial x_{n}}}={\frac {\partial v}{\partial y_{n}}}\\{\frac {\partial u}{\partial y_{n}}}=-{\frac {\partial v}{\partial x_{n}}}\end{array}}\right.$

Again, using Wirtinger derivatives this system of equation can be written in the following more compact form:

$\left\{{\begin{array}{l}{\frac {\partial f}{\partial {\bar {z_{1}}}}}=0\\\quad \quad \vdots \\{\frac {\partial f}{\partial {\bar {z_{n}}}}}=0\end{array}}\right.$

Notations for the case n>1

In the French, Italian and Russian literature on the subject, the multi-dimensional Cauchy-Riemann system is often identified with the following notation:

${\bar {\partial }}f$

The Anglo-Saxon literature (English and North American) uses the same symbol for the complex differential form related to the same operator.

References

• Hörmander, Lars (1990), An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3^rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, Zbl 0685.32001, ISBN 0-444-88446-7 ^[e].