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Contour integral

Evaluate the integral

    \[\mathcal{J}_{n, k} = \oint \limits_{\left | z \right |=1} \frac{\left ( 1+z \right )^n}{z^{k+1}} \, \mathrm{d}z\]

Solution

The function \displaystyle \frac{\left ( 1+z \right )^n}{z^{k+1}} is meromorphic on \mathbb{C}. Its only pole is 0 of order k+1. Hence,

    \begin{align*} \mathfrak{Res} \left ( f; z=0 \right ) &=\frac{1}{k!} \lim_{z \rightarrow 0} \left ( z^{k+1} f(z) \right )^{(k)} \\ &=\frac{1}{k!} \lim_{z\rightarrow 0} \left ( \left ( 1+z \right )^n \right )^{(k)} \\ &= \frac{n \left ( n-1 \right ) \cdots \left ( n- k+1 \right )}{k!} \\ &= \binom{n}{k} \end{align*}

Therefore,

    \[\mathcal{J}_{n, k} = 2 \pi i \binom{n}{k}\]

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