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 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.
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.
The subscripts are omitted when n=1.
The Cauchy-Riemann equations in ℂ (n=1)
Using Wirtinger derivatives these equation can be written in the following more compact form:
The Cauchy-Riemann equations in ℂn (n>1)
Again, using Wirtinger derivatives this system of equation can be written in the following more compact form:
Notations for the case n>1
- 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 .
- D'Alembert, Jean Le-Rond (1752), Essai d'une nouvelle théorie de la résistance des fluides, Paris: David (in French).
- Hörmander, Lars (1990), An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, Zbl 0685.32001, ISBN 0-444-88446-7 .
- Lamb, Sir Horace (1932), Hydrodynamics, Cambridge Mathematical Library (1995 paperback reprint of the 6th ed.), Cambridge: Cambridge University Press, Zbl 0828.01012, ISBN 0-521-45868-4 .