Radius of converge of Laurent series for $\frac{1}{\sin z}$

Question

I want to show that the radius of convergence of Laurent series for $\frac{1}{\sin z}$ is $\pi$

I showed that:

\begin{align} \frac 1 {\sin z} & = \frac 1 {z - \frac{z^3}{3!} + \frac{z^5}{5!} + \cdots } = \frac 1 {z\big(1 - (\frac{z^2}{3!} - \frac{z^4}{5!} + \cdots )\big)} \\[10pt] & = \frac{1}{z} \bigg( 1 + \bigg(\frac{z^2}{3!} - \frac{z^4}{5!} + \cdots \bigg) + \bigg(\frac{z^2}{3!} - \frac{z^4}{5!} + \cdots \bigg)^2 + \cdots \bigg) \\[10pt] & = \frac{1}{z} +\frac{z}{3!} + \bigg( \frac{1}{3!}- \frac{1}{5!}\bigg)z^3 + \cdots \end{align}

This equality holds if and only if: $$\bigg|\frac{z^2}{3!} - \frac{z^4}{5!} + \cdots \bigg| < 1 \iff \bigg|\frac{\sin z}{z} - 1\bigg| < 1$$

How can I show that the last inequality implies $|z| < \pi$ ?

Answer 1

After writing my comment, I realize that it is way simpler...apparently:

The function $\;\cfrac1{\sin z}\;$ is analytic at $\;\left\{\,z\in\Bbb C\;/\;0<|z|<\pi\,\right\}\;$ and thus the radius of convergence of (the non principal part of) the Laurent series of the function around zero is at least $\;\pi\;$ . But as already commented, it can't be $\;\ge\pi\;$ as the function has a pole of order one at $\;z=\pi\;$ . Thus , the radius is exactly $\;\pi\;$ .

Answer 2

The radius of convergence of the Laurent series of $1/\sin z$ about $z=0$ is $\pi$. The reason is that its poles are at $n\pi$, $n\in\Bbb Z$ (because that's where the sine has its zeroes). The nearest poles to zero are at $\pm \pi$, as distance $\pi$ from the origin.

Answer 3

Since $\text{Res}\left(\frac{1}{\sin z},z=k\pi\right) = (-1)^m $ for any $k\in\mathbb{Z}$, we have

$$ \frac{1}{\sin z} = \frac{1}{z}+\sum_{m\geq 1}(-1)^m\left(\frac{1}{z-m\pi}+\frac{1}{z+m\pi}\right) \tag{1}$$ over any compact subset of $\mathbb{C}\setminus\pi\mathbb{Z}$. In particular: $$ \frac{1}{\sin z}-\frac{1}{z} = 2z\sum_{m\geq 1}\frac{(-1)^{m+1}\frac{1}{m^2 \pi^2}}{1-\frac{z^2}{m^2\pi^2}}=2z\sum_{m\geq 1}z^{2m-2}\frac{\eta(2m)}{\pi^{2m}} \tag{2} $$ the radius of convergence of the Laurent series at the origin is $\pi$ (due to $\eta(2m)\approx 1$ for large values of $m$), as expected since $\pi$ is the distance from the origin of the closest singularities (simple poles at $z=\pm \pi$).