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 2n partial differential equations, where 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 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.

Historical note

The first introduction and use of the Cauchy-Riemann equations for n=1 is due to Jean Le-Rond D'Alembert in his 1752 work on hydrodynamics: this connection between complex analysis and hydrodynamics is made explicit in classical treatises of the latter subject, such as Horace Lamb's monumental work.

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

• Burckel, Robert B. (1979), An Introduction to Classical Complex Analysis. Vol. 1, Lehrbucher und Monographien aus dem Gebiete der exakten Wissenschaften. Mathematische Reihe, vol. Band 64, Basel–Stuttgart–New York–Tokyo: Birkhäuser Verlag, ISBN 3-7643-0989-X, at 570 ^[e].
• 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].