# Cauchy-Riemann equations

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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 spacen considered. Precisely, their homogeneous form express a necessary and sufficient condition between the real and imaginary part of a given complex valued function of 2n real variables 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[1]: this connection between complex analysis and hydrodynamics is made explicit in classical treatises of the latter subject, such as Horace Lamb's monumental work[2].

## 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

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

The subscripts are 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

{\displaystyle \left\{{\begin{aligned}{\frac {\partial u}{\partial x}}&={\frac {\partial v}{\partial y}}\\{\frac {\partial u}{\partial y}}&=-{\frac {\partial v}{\partial x}}\\\end{aligned}}\right.}

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

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

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

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

{\displaystyle \left\{{\begin{aligned}{\frac {\partial u}{\partial x_{1}}}&={\frac {\partial v}{\partial y_{1}}}\\{\frac {\partial u}{\partial y_{1}}}&=-{\frac {\partial v}{\partial x_{1}}}\\&\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{aligned}}\right.}

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

{\displaystyle \left\{{\begin{aligned}{\frac {\partial f}{\partial {\bar {z_{1}}}}}&=0\\&\vdots \\{\frac {\partial f}{\partial {\bar {z_{n}}}}}&=0\end{aligned}}\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:

${\displaystyle {\bar {\partial }}f}$

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