sum of values of holomorphic function

Let $w(z)$ be a holomorphic function on a simple closed curve $C$ and inside it.  If $a_1,a_2,\mathrm{\dots },a_m$ are inside $C$ the simple zeros of a function $f(z)$ holomorphic on $C$ and inside, then

$w(a_1)+w(a_2)+\mathrm{\dots }+w(a_m)={\displaystyle \frac{1}{2i\pi }}{\displaystyle {\oint }_C}w(z){\displaystyle \frac{f^{\prime }(z)}{f(z)}}𝑑z$ (1)

where the contour integral is taken anticlockwise.

The if some of the zeros are multiple and are counted with multiplicities (http://planetmath.org/Pole).

If the zeros $a_j$ of $f(z)$ have the multiplicities ${\alpha }_j$ and the function has inside $C$ also the poles $b_1,b_2,\mathrm{\dots },b_n$ with the multiplicities ${\beta }_1,{\beta }_2,\mathrm{\dots },{\beta }_n$,  then (1) must be written

${\displaystyle \sum _j}{\alpha }_{j}w(a_j)-{\displaystyle \sum _k}{\beta }_{k}w(b_k)={\displaystyle \frac{1}{2i\pi }}{\displaystyle {\oint }_C}w(z){\displaystyle \frac{f^{\prime }(z)}{f(z)}}𝑑z.$ (2)

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

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