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单词 ZeroAsContourIntegral
释义

zero as contour integral


Suppose that f is a complex function which is defined in some openset D which has a simple zero at some pointpD. Then we have

p=12πiCzf(z)f(z)𝑑z

where C is a closed path in D which encloses p but does not enclose orpass through any other zeros of f.

This follows from the Cauchy residue theorem. We have that the poles off/f occur at the zeros of f and that the residueMathworldPlanetmath of a pole off/f is 1 at a simple zero of f. Hence, the residue ofzf(z)/f(z) at p is p, so the above follows from the residue theorem.

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更新时间:2025/5/4 13:08:46