path-C-is-closed-f-is-regular-function-in-Path-C-f-is-have-zero-point-and-poles-in-C-show-that-C-f-z-f-z-dz-2pii-Z-P-and-if-poles-are-not-exist-show-that-C-f-z-f-z-dz-2

Question Number 213846 by issac last updated on 18/Nov/24

$$\mathrm{path}\:\mathscr{C}\:\mathrm{is}\:\mathrm{closed} \\ $$$${f}\:\mathrm{is}\:\mathrm{regular}\:\mathrm{function}\:\mathrm{in}\:\mathrm{Path}\:\mathscr{C} \\ $$$${f}\:\mathrm{is}\:\mathrm{have}\:\mathrm{zero}\:\mathrm{point}\:\mathrm{and}\:\mathrm{poles}\:\mathrm{in}\:\mathscr{C} \\ $$$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\oint_{\:\mathscr{C}} \:\frac{{f}\left({z}\right)}{{f}'\left({z}\right)}\:\mathrm{d}{z}=\mathrm{2}\pi\boldsymbol{{i}}\left({Z}−{P}\right) \\ $$$$\mathrm{and}\:\mathrm{if}\:\mathrm{poles}\:\mathrm{are}\:\mathrm{not}\:\mathrm{exist} \\ $$$$\mathrm{show}\:\mathrm{that} \\ $$$$\oint_{\:\mathscr{C}\:} \:\frac{{f}\left({z}\right)}{{f}'\left({z}\right)}\:\mathrm{d}{z}=\mathrm{2}\pi\boldsymbol{{i}}{Z} \\ $$$$\mathrm{this}\:\mathrm{equation}\:\mathrm{is}\:\mathrm{equivalent}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{formula}\:\mathrm{for}\:\mathrm{finding}\:\mathrm{the}\:\mathrm{numbef}\:\mathrm{of}\:\mathrm{solution} \\ $$

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