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| − | ==HELLO GUYS==
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| − | こんにちは！
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| − | Hello
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| − | <math>f(z)=\sum_{j=-\infty}^{\infty} C_n (z-\alpha)^n</math>
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| − | <math>C_n=\frac{1}{2\pi i}\int\frac{f(\xi)}{(\xi-\alpha)^{n+1}}</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\oint_{\gamma_k}f(z)dz</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\oint_{\gamma_k}\sum_{j=-\infty}^{\infty}C_n (z-\alpha_k)^n</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\sum_{j=-\infty}^{\infty}C_n\oint_{\gamma_k} (z-\alpha_k)^n</math>
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| − | <math>z(\theta)=\alpha_k+ae^{i\theta}</math> <math>(0\leq\theta\leq 2\pi)</math>
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| − | <math>dz=iae^{i\theta}d\theta</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\sum_{j=-\infty}^{\infty}C_n\int_{0}^{2\pi} (ae^{i\theta})^j iae^{i\theta}d\theta</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\sum_{j=-\infty}^{\infty}C_n ia^{j+1} \int_{0}^{2\pi}e^{i(j+1)\theta}d\theta</math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}\sum_{j=-\infty}^{\infty}C_n ia^{j+1} [\frac{1}{i(n+1)}e^{i(j+1)\theta}]_{0}^{2\pi}</math>
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| − | <math> \int_{0}^{2\pi}e^{i(n+1)\theta}d\theta =\begin{cases}0 & n\neq -1\\2\pi i & n=-1\end{cases} </math>
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| − | <math>\oint_{\gamma}f(z)dz = \sum_{k=1}^{n}2\pi i C_{-1}</math>
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| − | <cmath>\boxed{\oint_{\gamma}f(z)dz = 2\pi i\sum_{k=1}^{n}res(f(z),\alpha_{k})}</cmath>
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| − | ==contributions==
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| − | [[2016 AIME I Problems/Problem 10]] Solution 4
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