Example 7. Compute the integral

$$
I=\int_{|z|=3} \frac{z^{17}}{\left(z^{2}+2\right)^{3}\left(z^{3}+3\right)^{4}} d z
$$
Solution. The integrand function

$$
f(z)=\frac{z^{17}}{\left(z^{2}+2\right)^{3}\left(z^{3}+3\right)^{4}}
$$

inside the contour $|z|=3$ has five singular points, which are multiple poles. Using the main theorem of residues leads to extensive calculations. For the calculation of this integral, it is more convenient to use equation (3), according to which we will have

$$
I=-2 \pi i \text{ res } f(\infty)
$$

Since the function $f(z)$ can be represented as

$$
f(z)=\frac{z^{17}}{\left(z^{2}+2\right)^{3}\left(z^{3}+3\right)^{4}}=\frac{z^{17}}{z^{6}\left(1+\frac{2}{z^{2}}\right)^{3}\left(1+\frac{3}{z^{3}}\right)^{4} z^{12}}=\frac{1}{z} \cdot \frac{1}{\left(1+\frac{2}{z^{2}}\right)^{3}\left(1+\frac{3}{z^{3}}\right)^{4}}
$$

it is clear from this that the principal part of the Laurent series expansion of this function in the neighborhood of the infinitely distant point $z=\infty$ starts with the term $1 / z$. Therefore, res $f(\infty)=-1$. Substituting this value into equation (3), we get $I=2 \pi i$.

## Problems for Independent Solution

Determine the nature of the infinitely distant point for the following functions:

366. $f(z)=\frac{z^{3}-z^{2}+z+6}{z^{2}}$. 367. $f(z)=\frac{z+1}{z^{4}}$.
367. $f(z)=\frac{e^{z}}{z^{2}}$. 369. $f(z)=\cos \frac{1}{z} \cdot$ 370. $f(z)=e^{1 / z^{2}} \cdot$ 371. $f(z)=z^{3} e^{1 / z}$.
368. Let the function $f(z)$ be representable in the form $f(z)=\varphi\left(\frac{1}{z}\right)$, where the function $\varphi(\zeta)$ is analytic at the point $\zeta=0$. Prove that res $f(z)=-\varphi^{\prime}(0)$.

Using the residue at the infinitely distant point, compute the following integrals:

373. $\int_{|z|=1} \frac{z^{2}+1}{z^{3}} d z$.
374. $\int_{|z|=2} \frac{d z}{1+z^{12}}$. 377. $\int_{|z|=1} z^{2} \sin \frac{1}{z} d z$.
375. $\int_{|z|=2} \frac{1000 z+2}{1+z^{1224}} d z$.
376. $\int_{|z|=3} \frac{z^{9}}{z^{10}-1} d z$.
377. $\int_{|z|=3} \frac{e^{x}}{z-1} d z$.

$2^{\circ}$. Application of residues to the computation of definite integrals. 1. Integrals of rational functions. Let $f(x)$ be a rational function, $f(x)=\frac{P_{m}(x)}{Q_{n}(x)}$, where $P_{m}(x)$ and $Q_{n}(x)$ are polynomials of degrees $m$ and $n$, respectively. If $f(x)$ is continuous on the entire real axis $\left(Q_{n}(x) \neq 0\right)$ and $n \geqslant m+2$, i.e., the degree of the denominator is at least two units greater than the degree of the numerator, then

$$
\int_{-\infty}^{+\infty} f(x) d x=2 \pi i \sigma
$$

where $\sigma$ denotes the sum of the residues of the function $f(z)=\frac{P_{m}(z)}{Q_{n}(z)}$ at all poles located in the upper half-plane.