Example 9. Find the residue of the function

$$
f(z)=e^{1 / z^{2}} \cos z
$$

at the point $z=0$.
Solution. Since the residue at the point $z=0$ is equal to the coefficient of $z^{-1}$, we immediately obtain that in this case the residue is zero, since the function $f(z)$ is even and its expansion in the neighborhood of the point $z=0$ cannot contain odd powers of $z$.

## Problems for Independent Solution

Find the residues at the singular points of the following functions:

324. $f(z)=\frac{\tan z}{z^{2}-\frac{\pi}{4} z}$.
325. $f(z)=z^{3} e^{1 / x}$.
326. $f(z)=\frac{\cosh z}{\left(z^{2}+1\right)(z-3)}$.
327. $f(z)=\frac{e^{z}}{\frac{1}{4}-\sin ^{2} z}$.
328. $f(z)=\frac{e^{z}}{z^{3}(z-1)}$.
329. $f(z)=\frac{z}{(z+1)^{3}(z-2)^{2}}$.
330. $f(z)=\frac{e^{-1 / z^{2}}}{1+z^{4}}$.
331. $f(z)=z^{2} \sin \frac{1}{z}$.
332. $f(z)=\cos \frac{1}{z}+z^{3}$.
333. $f(z)=\frac{\sin 2 z}{(z+i)\left(z-\frac{i}{2}\right)^{2}}$.
334. $f(z)=\frac{1-\cos z}{z^{3}(z-3)}$.
335. $f(z)=e^{z^{2}+1 / z^{2}}$.
336. $f(z)=\frac{e^{i z}}{\left(z^{2}-1\right)(z+3)}$.
337. $f(z)=\frac{\cos z}{z^{3}-\frac{\pi}{2} z^{2}}$.
338. $f(z)=\frac{e^{\pi z}}{z-i}$.
339. $f(z)=\frac{z^{2 n}}{(z-1)^{n}} \quad(n>0-$ integer).
340. $f(z)=\cot ^{2} z$.
341. $f(z)=\sin z \cos \frac{1}{z}$.
342. $f(z)=e^{z /(z-1)}$.
343. $f(z)=\frac{\sin \frac{1}{z}}{1-z}$.
344. $f(z)=\frac{e^{1 / z}}{1+z}$.
345. $f(z)=e^{\left(z^{2}+1\right) / z}$ 346. $f(z)=e^{z} \sin \frac{1}{z}$.

## § 11. The Residue Theorem of Cauchy. Application of Residues to the Evaluation of Definite Integrals. Summation of Some Series Using Residues

## $1^{\circ}$. The Residue Theorem of Cauchy.

Theorem. If the function $f(z)$ is analytic on the boundary $C$ of the region $D$ and everywhere inside the region, except for a finite number of singular points $z_{1}, z_{2}, \ldots, z_{n}$, then

$$
\int_{C} f(z) d z=2 \pi i \sum_{k=1}^{n} \operatorname{res} f\left(z_{k}\right)
$$