Math Problems Vault

Problem 14.1

Compute the integrals of rational functions:

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(1)

$\oint_{\left|z\right| = 1} \frac{d z}{z}$ ;

(2)

$\oint_{\left|z\right| = 2} \frac{d z}{(z-1)^{2}}$ ;

(3)

$\oint_{\left|z\right| = 2} \frac{d z}{z^{2}+1}$ ;

(4)

$\oint_{\left|z\right| = 3} \frac{d z}{z^{2}+z+1}$ ;

(5)

$\oint_{\left|z\right| = 3} \frac{z^{4}}{z^{2}+z-2} d z$ ;

(6)

$\oint_{\left|z+i\right| = 2} \frac{d z}{z^{2}\left(z^{2}-7 z+12\right)}$ .

Problem 14.2

Compute the integrals:

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(1)

$\oint_{\left|z\right| = 3} \frac{z \cos \frac{\pi }{z}}{z+2 i} d z$ ;

(2)

$\oint_{\left|z\right| = 4} \frac{e^{1 /(z-1)}}{z-2} d z$ ;

(3)

$\oint_{\left|z-\frac{1}{2}\right| = 1} \frac{z^{3} e^{1 / z}}{1-z^{2}} d z$ ;

(4)

$\oint_{\left|z\right| = 2} z^{2} \sin \frac{1}{1-z} d z$ ;

(5)

$\oint_{\left|z\right| = 1} \frac{1}{4 i+3 z} \cos \frac{5 i}{6 z+3 i} d z$ ;

(6)

$\oint_{\left|z\right| = 2} \frac{1}{3+i z} e^{4 i /(z-i)} d z$ ;

(7)

$\oint_{\left|z\right| = 3 / 2} \frac{z \operatorname {tg} z}{\left(z^{2}-1\right)^{2}} d z$ ;

(8)

$\oint_{\left|z\right| = 5 / 2} \frac{z^{2}}{z-3} \sin \frac{z}{z-2} d z$ ;

(9)

$\oint_{\left|z-i\right| = 3} \frac{z}{z^{2}+9} \operatorname {ch} \frac{z}{z-2} d z$ ;

(10)

$\oint_{\left|z\right| = 6} \frac{z^{2}}{e^{i z}+i} d z$ ;

(11)

$\oint_{\left|z\right| = 3} \frac{d z}{(z-1)^{2}\left(1-e^{2 z}\right)}$ ;

(12)

$\oint_{\left|z\right| = 2} \frac{z^{2}}{z-1} e^{1 /(z-1)} d z$ ;

(13)

$\oint_{\left|z+\frac{1}{3}\right| = \frac{1}{6}} \frac{3 z+1}{1-\sin \frac{\pi }{2 z}} d z$ ;

(14)

$\oint_{\left|z\right| = 4} \frac{z d z}{e^{1 / z}+e^{1 / 2 z}}$ ;

(15)

$\oint_{\left|z\right| = 2} \frac{e^{z^{5}}}{z\left(1+z^{5}\right)} d z$ ;

(16)

$\oint_{\left|z-\frac{\pi }{2}(1-i)\right| = \pi } \frac{z d z}{\cos z-\operatorname {ch} z}$ ;

(17)

$\oint_{\left|z\right| = 1} \frac{(z-i) \sin \frac{1}{i z}}{(z-3 i)^{2}} d z$ ;

(18)

$\oint_{\left|z\right| = 1} \frac{(z+1) \cos \frac{i}{z}}{(2 i-z)^{2}} d z$ ;

(19)

$\oint_{\left|z\right| = 1 / 2} \frac{z+\frac{2}{\pi }}{\left(2 \sin \frac{1}{3 z}-1\right)} d z$ ;

(20)

$\oint_{\left|z+i\right| = 3 / 2} \frac{2(z+3 i) \operatorname {ch} \frac{1}{2 z}}{z^{4}+10 z^{2}+9} d z$ ;

(21)

$\oint_{\left|z\right| = 1} \frac{z-\frac{1}{\ln 2}}{e^{1 / z}-2} d z$ ;

(22)

$\oint_{\left|z-\frac{\pi }{2}\right| = 3} \frac{2 z-i \ln 2}{\operatorname {tg} z-\frac{i}{3}} d z$ ;

(23)

$\oint_{\left|z+1-i\right| = 2} \frac{z+i}{(z-i) \operatorname {sh} \frac{1}{2 z}} d z$ ;

(24)

$\oint_{\left|z-1\right| = 1} \frac{z d z}{(\pi -3 z)(1+\cos 3 z)}$ .

Problem 14.3

(1)(2)(3)(4)

Making an appropriate change of variable, reduce the given integral to an integral over a closed contour in $\mathbb {C}$ and compute it:

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(1)

$\int_{0}^{2 \pi } \frac{d \varphi }{a+\cos \varphi }(a > 1)$ ;

(2)

$\int_{0}^{2 \pi } \frac{d \varphi }{(a+b \cos \varphi )^{2}} \quad (a > b > 0)$ ;

(3)

$\int_{0}^{2 \pi } \frac{d \varphi }{\left(a+b \cos^{2} \varphi \right)^{2}}$ ;

(4)

$\int_{0}^{2 \pi } e^{\cos \varphi } \cos (n \varphi -\sin \varphi ) d \varphi , \quad n \in \mathbb {Z}$ .

Remark.

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In Problems 4--6 it is assumed that, while moving along the boundary $\Gamma$ of the domain $D$ in the direction of integration, the domain $D$ itself remains on the left.

Problem 14.4

Prove that the integral

\int _{\Gamma } \frac{e^{i z}}{(z-i)^{2}} d z

taken over the boundary $\Gamma$ of the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ , equals the sum of the residues of the integrand in this half-plane, and find its value.

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Problem 14.5

(1)(2)(3)

Verify the applicability of the residue theorem to the following integrals, taken over the boundaries $\Gamma$ of unbounded domains $D$ , and compute these integrals:

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(1)

$\int_{\Gamma } \frac{z e^{-z}}{z^{2}-1} d z \quad (D = \left\{ z: \mathrm{Re}\left(z\right) > 0\right\} )$ ;

(2)

$\int_{\Gamma } \frac{e^{z}}{\operatorname {sh} 2 z} d z \quad \left(D = \left\{ z:-\frac{\pi }{4} < \mathrm{Im}\left(z\right) < \frac{\pi }{4}\right\} \right)$ ;

(3)

$\int_{\Gamma } \frac{z^{3}}{(z-1)^{2}} e^{-z^{3}} d z \quad \left(D = \left\{ z:-\frac{\pi }{6} < \arg z < \frac{\pi }{6}\right\} \right)$ .

Problem 14.6

(1)(2)

Verify that the residue theorem is not applicable to the integrals taken over the boundaries $\Gamma$ of the indicated domains $D$ :

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(1)

$I = \int_{\Gamma } e^{-z^{2}} d z \quad (D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} )$ ;

(2)

$I = \int_{\Gamma } \frac{\sin z}{1+z^{2}} d z \quad (D = \left\{ z: \mathrm{Im}\left(z\right) > 0\right\} )$ .

Problem 14.7

Let the functions $f(z)$ and $g(z)$ be regular in the closure $\bar{D}$ of a bounded domain $D$ , where the function $f(z)$ has, in the domain $D$ , zeros $a_{1}, \ldots , a_{n}$ (each zero written as many times as its order) and has no other zeros either in the domain $D$ or on its boundary $\Gamma$ . Prove that

\oint _{\Gamma } g(z) \frac{f^{\prime }(z)}{f(z)} d z = 2 \pi i \sum _{k = 1}^{n} g\left(a_{k}\right)

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Problem 14.8

Let the function $f(z)$ of Problem 7 also have, in the domain $D$ , poles $b_{1}, \ldots , b_{m}$ (each pole also written as many times as its order). Prove that

\oint _{\Gamma } g(z) \frac{f^{\prime }(z)}{f(z)} d z = 2 \pi i \sum _{k = 1}^{n} g\left(a_{k}\right)-2 \pi i \sum _{k = 1}^{m} g\left(b_{k}\right)

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Problem 14.9

Let the function $f(z)$ be regular in the closure $\bar{D}$ of a bounded domain $D$ with boundary $\Gamma$ , and let the points $a_{1}, \ldots , a_{n}$ lie in the domain $D$ and be pairwise distinct. Denote

P(z) = \left(z-a_{1}\right) \ldots \left(z-a_{n}\right)

and

\Phi (z) = -\frac{P(z)}{2 \pi i} \oint _{\Gamma } \frac{f(\xi )}{P(\xi )} \frac{d \xi }{\xi -z} \quad (z \notin \bar{D})

Prove that the function $\Phi (z)$ extends analytically to the entire plane and is a polynomial of degree $n-1$ satisfying the conditions

\Phi \left(a_{k}\right) = f\left(a_{k}\right) \quad (k = 1,2, \ldots , n).

(The polynomial $\Phi (z)$ is called the Lagrange interpolation polynomial.)

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Problem 14.10

Prove the formula

\frac{1}{2 \pi i} \oint _{\left|z\right| = 1} e^{\frac{x}{2}\left(z+\frac{1}{z}\right)} d z = \sum _{n = 0}^{\infty }\left(\frac{x}{2}\right)^{2 n+1} \frac{1}{n!(n+1)!}.

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