# 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 ℂ* ^{n}* 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 ℂ^{n}≡ℝ^{2n}, identifying the points of the euclidean spaces on the complex and real fields as follows

The subscripts are omitted when `n`=1.

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

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

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

### The Cauchy-Riemann equations in ℂ^{n} (`n`>1)

^{n}

Let `f`(`x _{1}`,

`y`,...,

_{1}`x`,

_{n}`y`) =

_{n}`u`(

`x`,

_{1}`y`,...,

_{1}`x`,

_{n}`y`) +

_{n}`i`

`v`(

`x`,

_{1}`y`,...,

_{1}`x`,

_{n}`y`) a complex valued differentiable function. Then

_{n}`f`satisfies the homogeneous Cauchy-Riemann equations if and only if

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

### 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:

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

## Notes

- ↑ See D'Alembert 1752.
- ↑ See Lamb 1932.

## 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. 64, Basel–Stuttgart–New York–Tokyo: Birkhäuser Verlag, ISBN 3-7643-0989-X^{[e]}. - D'Alembert, Jean Le-Rond (1752),
*Essai d'une nouvelle théorie de la résistance des fluides*, Paris: David^{[e]}(in French). - 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]}. - Lamb, Sir Horace (1932),
*Hydrodynamics*, Cambridge Mathematical Library (1995 paperback reprint of the 6^{th}ed.), Cambridge: Cambridge University Press, Zbl 0828.01012, ISBN 0-521-45868-4^{[e]}.