Example 10. Determine the nature of the singular point $z=1$ of the function

$$
f(z)=\frac{\sin \pi z}{2 e^{z-1}-z^{2}-1}
$$
Solution. Consider the function

$$
\varphi(z)=\frac{1}{f(z)}=\frac{2 e^{z-1}-z^{2}-1}{\sin \pi z}
$$

The point $z=1$ is a zero of the third order for the numerator

$$
\psi(z)=2 e^{z-1}-z^{2}-1
$$

since

$$
\begin{gathered}
\psi(1)=0 ; \quad \psi^{\prime}(1)=\left.\left(2 e^{z-1}-2 z\right)\right|_{z=1}=0 ; \\
\psi^{\prime \prime}(1)=\left.\left(2 e^{z-1}-2\right)\right|_{z=1}=0 ; \quad \psi^{\prime \prime \prime}(1)=\left.2 e^{z-1}\right|_{z=1}=2 \neq 0
\end{gathered}
$$

The point $z=1$ is a zero of the first order for the denominator $\sin \pi z$ of the function $\varphi(z)$.

Therefore, the point $z=1$ will be a zero of order $3-1=2$ for the function $\varphi(z)$, and thus a pole of the second order for the given function.

## Problems for Independent Solution

Determine the nature of the singular point $z_{0}=0$ for the following functions:
300. a) $\frac{1}{z-\sin z}$
b) $\frac{1}{\cos z-1+\frac{1}{2} z^{2}}$
c) $\frac{1}{e^{-z}+z-1}$

301. a) $\frac{\sin z}{e^{-z}+z-1}$; b) $\frac{\operatorname{sh} z}{z-\operatorname{sh} z}$.

Find the singular points and determine their nature for the following functions:
302. a) $\frac{1}{1-\sin z}$;
b) $\frac{1-\cos z}{z^{2}}$
303. a) $e^{1 /(x+2)}$;
b) $\cos \frac{1}{z}$

304. a) $\frac{z}{z^{5}+2 z^{4}+z^{3}}$

b) $\frac{1}{e^{-z}-1}+\frac{1}{z^{2}}$.

305. a) $e^{-1 / z^{2}}$

b) $\sin \frac{\pi}{z+1}$
c) $\operatorname{ch} \frac{1}{z}$

306. a) $\frac{z^{2}}{\cos z-1}$

b) $\frac{1-\sin z}{\cos z}$

c) $\frac{z-\pi}{\sin ^{2} z}$

The following statements are true.

1. For a point $z_{0}$ to be a removable singular point of the function $f(z)$, it is necessary and sufficient that the Laurent series expansion of $f(z)$ in a neighborhood of the point $z_{0}$ does not contain the principal part.
2. For a point $z_{0}$ to be a pole of the function $f(z)$, it is necessary and sufficient that the principal part of the Laurent series expansion of $f(z)$ in a neighborhood of $z_{0}$ contains only a finite number of terms

$$
f(z)=\frac{c_{-k}}{\left(z-z_{0}\right)^{k}}+\ldots+\frac{c_{-1}}{z-z_{0}}+\sum_{n=0}^{\infty} c_{n}\left(z-z_{0}\right)^{n} \quad\left(c_{-k} \neq 0\right)
$$

The largest of the exponents of the powers of the differences $z-z_{0}$ contained in the denominators of the terms of the principal part of the Laurent series coincides with the order of the pole.

3. A point $z_{0}$ is a singular essential point for the function $f(z)$ if and only if the principal part of its Laurent series expansion in a neighborhood of the point $z_{0}$ contains infinitely many terms.