holomorphic functions of several variables


Let Ωn be a domain and let f:Ω be a function. f is called if it is holomorphic (http://planetmath.org/Holomorphic) in each variable separately as a function of one variable.

That means that the function zkf(z1,,zk,,zn) is holomorphic as a function of one variable. It is not at all obvious that such a function is even continuousMathworldPlanetmath and we must apply the Hartogs’s theorem on separate analyticity which is not a trivial result.

Historically and some authors today still continue to do so, the definition of being holomorphic in several variables did include the continuity or at least local boundedness requirement.

Of course we can also characterize holomorphic functions by their power seriesMathworldPlanetmath.


f is holomorphic in Ω if and only if near each point ζΩ there is a neighbourhood U and a power series in several variables


where α ranges over all the multi-indices, aαC and such that the series converges to f(z) for zU.

Another way to characterize holomorphic functions is by the use of the Cauchy-Riemann equationsMathworldPlanetmath, which can be given in a very form by the ¯-operator (http://planetmath.org/BarpartialOperator).


f is holomorphic if and only if ¯f=0.

Despite the similaritiesMathworldPlanetmath, one should be careful about carelessly generalizing results about functions of one variable to functions of several variables as the theory is quite different. See the topic entry on several complex variables for more .


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title holomorphic functions of several variables
Canonical name HolomorphicFunctionsOfSeveralVariables
Date of creation 2013-03-22 15:33:42
Last modified on 2013-03-22 15:33:42
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32A10