sum of values of holomorphic function

Let w(z) be a holomorphic functionMathworldPlanetmath on a simple closed curve C and inside it.  If a1,a2,,am are inside C the simple zeros of a function f(z) holomorphic on C and inside, then

w(a1)+w(a2)++w(am)=12iπCw(z)f(z)f(z)𝑑z (1)

where the contour integral is taken anticlockwise.

The if some of the zeros are multipleMathworldPlanetmath and are counted with multiplicities (

If the zeros aj of f(z) have the multiplicities αj and the function has inside C also the poles b1,b2,,bn with the multiplicities β1,β2,,βn,  then (1) must be written

jαjw(aj)-kβkw(bk)=12iπCw(z)f(z)f(z)𝑑z. (2)

The special case  w(z)1  gives from (2) the argument principle.


  • 1 Ernst Lindelöf: Le calcul des résidus et ses applications à la théorie des fonctions.  Gauthier-Villars, Paris (1905).
Title sum of values of holomorphic function
Canonical name SumOfValuesOfHolomorphicFunction
Date of creation 2013-03-22 19:15:30
Last modified on 2013-03-22 19:15:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 30E20