Cauchy integral formula in several variables

Let D=D1××Dnn be a polydisc.


Let f be a function continuous in D¯ (the closure of D). Then f is holomorphic ( in D if and only if for all z=(z1,,zn)D we have


As in the case of one variable this theorem can be in fact used as a definition of holomorphicity. Note that when n>1 then we are no longer integrating over the boundary of the polydisc but over the distinguished boundary, that is over D1××Dn.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Cauchy integral formula in several variables
Canonical name CauchyIntegralFormulaInSeveralVariables
Date of creation 2013-03-22 15:33:46
Last modified on 2013-03-22 15:33:46
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 32A07
Classification msc 32A10