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

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

$$\mathrm{path}\mathcal{C}\mathrm{is}\mathrm{closed}$$$$f\mathrm{is}\mathrm{regular}\mathrm{function}\mathrm{in}\mathrm{Path}\mathcal{C}$$$$f\mathrm{is}\mathrm{have}\mathrm{zero}\mathrm{point}\mathrm{and}\mathrm{poles}\mathrm{in}\mathcal{C}$$$$\mathrm{show}\mathrm{that}$$$${\oint }_{\mathcal{C}}\frac{f\left(z\right)}{f^{\prime }\left(z\right)}\mathrm{d}z=2\pi \mathit{i}(Z-P)$$$$\mathrm{and}\mathrm{if}\mathrm{poles}\mathrm{are}\mathrm{not}\mathrm{exist}$$$$\mathrm{show}\mathrm{that}$$$${\oint }_{\mathcal{C}}\frac{f\left(z\right)}{f^{\prime }\left(z\right)}\mathrm{d}z=2\pi \mathit{i}Z$$$$\mathrm{this}\mathrm{equation}\mathrm{is}\mathrm{equivalent}\mathrm{to}\mathrm{the}$$$$\mathrm{formula}\mathrm{for}\mathrm{finding}\mathrm{the}\mathrm{numbef}\mathrm{of}\mathrm{solution}$$

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