# sum of values of holomorphic function

Let $w(z)$ be a holomorphic function on a simple closed curve $C$ and inside it.  If $a_{1},\,a_{2},\,\ldots,\,a_{m}$ are inside $C$ the simple zeros of a function $f(z)$ holomorphic on $C$ and inside, then

 $\displaystyle w(a_{1})\!+\!w(a_{2})\!+\ldots+\!w(a_{m})\;=\;\frac{1}{2i\pi}\!% \oint_{C}\!w(z)\frac{f^{\prime}(z)}{f(z)}\,dz$ (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 $b_{1},\,b_{2},\,\ldots,\,b_{n}$ with the multiplicities $\beta_{1},\,\beta_{2},\,\ldots,\,\beta_{n}$,  then (1) must be written

 $\displaystyle\sum_{j}\alpha_{j}w(a_{j})-\sum_{k}\beta_{k}w(b_{k})\;=\;\frac{1}% {2i\pi}\!\oint_{C}\!w(z)\frac{f^{\prime}(z)}{f(z)}\,dz.$ (2)

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

## References

• 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 SumOfValuesOfHolomorphicFunction 2013-03-22 19:15:30 2013-03-22 19:15:30 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 30E20