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A contour integral

Define

\displaystyle f(z) = \frac{1}{z} \cdot \frac{1-2z}{z-2} \cdots \frac{1-10z}{z-10}

Evaluate the contour integral \displaystyle \ointctrclockwise \limits_{|z|=100} f(z) \, {\rm d}z.

Solution

We are applying the substitution u=\frac{1}{z} thus:

    \begin{align*} \ointctrclockwise \limits_{\left | z \right |=100} f(z) \, {\rm d}z &=- \ointclockwise \limits_{\left | w \right |=1/100} f \left ( \frac{1}{w} \right ) \frac{{\rm d}w}{w^2} \\ &=\ointctrclockwise \limits_{\left | w \right |=1/100} \frac{1}{w} \prod_{n=1}^{5} \frac{w-2n}{1-2nw} \, {\rm d}w \\ &=- 2\pi i 3840 \end{align*}

since the function \displaystyle g(w) = \frac{1}{w} \prod_{n=1}^{5} \frac{w-2n}{1-2nw} has only one pole in the specific contour , namely w=0.

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