Prove that cot z = 1/z + 2z{1/z² - π2 + 1/z² - 4π² + ···}.

Question

Prove that [latex]\cot z=\frac{1}{z}+2 z\left\{\frac{1}{z^{2}-\pi^{2}}+\frac{1}{z^{2}-4 \pi^{2}}+\cdots\right\}[/latex].

Accepted Answer

We can write the result of Problem 7.34 in the form

[latex] \begin{aligned} \cot z & =\frac{1}{z}+\underset{N ightarrow \infty}{\lim}\left\{\sum_{n=-N}^{-1}\left(\frac{1}{z-n \pi}+\frac{1}{n \pi} ight)+\sum_{n=1}^{N}\left(\frac{1}{z-n \pi}+\frac{1}{n \pi} ight) ight\} \ & =\frac{1}{z}+\underset{N ightarrow \infty}{\lim}\left\{\left(\frac{1}{z+\pi}+\frac{1}{z-\pi} ight)+\left(\frac{1}{z+2 \pi}+\frac{1}{z-2 \pi} ight)+\cdots+\left(\frac{1}{z+N \pi}+\frac{1}{z-N \pi} ight) ight\} \ & =\frac{1}{z}+\underset{N ightarrow \infty}{\lim}\left\{\frac{2 z}{z^{2}-\pi^{2}}+\frac{2 z}{z^{2}-4 \pi^{2}}+\cdots+\frac{2 z}{z^{2}-N^{2} \pi^{2}} ight\} \ & =\frac{1}{z}+2 z\left\{\frac{1}{z^{2}-\pi^{2}}+\frac{1}{z^{2}-4 \pi^{2}}+\cdots ight\} \end{aligned} [/latex]

Question 7.35: Prove that cot z = 1/z + 2z{1/z² - π2 + 1/z² - 4π² + ···}.

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