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单词 SumOfValuesOfHolomorphicFunction
释义

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 (http://planetmath.org/Pole).

If the zeros aj of f(z) have the multiplicities αj and the function has inside C also the polesb1,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.

References

  • 1 Ernst Lindelöf: Le calcul des résidus et ses applications à la théorie des fonctions.  Gauthier-Villars, Paris (1905).
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更新时间:2025/5/5 1:56:09