Evaluating a contour integral of a rational or meromorphic function

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Problem 5.6.38 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

Let $f(z)$ be a meromorphic function on the complex plane. Suppose that for every polynomial $p(z) \in \mathbb {C}^{z}$ and every closed contour $\Gamma$ avoiding the poles of $f$ , we have

\int _{\Gamma } p(z)^{2}f(z) \, dz = 0.

Prove that $f(z)$ is entire.

Problem 5.10.6 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.10 "Residue Theory"

Prove that for any fixed complex number $\zeta$ ,

\sum _{n=0}^{\infty } \frac{1}{(n!)^{2}} \left(\frac{\zeta }{2}\right)^{2n} = \frac{1}{2\pi } \int _{0}^{2\pi } e^{2\zeta \cos \theta } \, d\theta .

Problem 5.10.10 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.10 "Residue Theory"

Let $C$ denote the positively oriented circle $\left|z\right| = 2$ , $z \in \mathbb {C}$ . Evaluate the integral

\int _{C} \sqrt{z^{2}-1} \, dz

where the branch of the square root is chosen so that $\sqrt{4} - 1 > 0$ .

Problem 5.10.14 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.10 "Residue Theory"

(1)(2)

(1)

Show that there is a unique analytic branch outside the unit circle of the function $f(z) = \sqrt{z^{2}+z+1}$ such that $f(t)$ is positive when $t > 1$ .

(2)

Using the branch determined in Part 1, calculate the integral

\frac{1}{2\pi i} \int _{C_{r}} \frac{dz}{\sqrt{z^{2}+z+1}}

where $C_{r}$ is the positively oriented circle $\left|z\right| = r$ and $r > 1$ .

Problem 5.10.18 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.10 "Residue Theory"

Evaluate

\int _{C} (2z-1) e^{z/(z-1)} \, dz

where $C$ is the circle $\left|z\right| = 2$ with counterclockwise orientation.

Problem 5.11.4 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.11 "Integrals Along the Real Axis"

Evaluate the integrals

F(t) = \int _{-\infty }^{\infty } \frac{e^{-itx}}{(x+i)^{3}} \, dx , \qquad -\infty < t < \infty .

Problem 5.11.6 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.11 "Integrals Along the Real Axis"

Evaluate

\int _{-\infty }^{\infty } \frac{\sin ^{3}x}{x^{3}} \, dx .

Problem 5.11.12 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.11 "Integrals Along the Real Axis"

Evaluate

\int _{-\infty }^{\infty } \frac{\cos x}{(1+x^{2})^{3}} \, dx .

Problem 5.11.19 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.11 "Integrals Along the Real Axis"

Evaluate

\int _{-\infty }^{\infty } \frac{x-\sin x}{x^{3}} \, dx .

Problem 5.11.20 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.11 "Integrals Along the Real Axis"

Evaluate

\int _{-\infty }^{\infty } \frac{dx}{(1+x+x^{2})^{2}} .

Problem 5.3.1 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.3 "Complex Integration"

Evaluate the integral

\int _{\left|z\right|=1} e^{e^z}\, dz.

Problem 5.3.2 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.3 "Complex Integration"

Evaluate

\int _\gamma \frac{dz}{\sin ^3 z},

where $\gamma$ is the positively oriented circle $\left\{ \left|z\right| = 1\right\}$ .

Problem 5427 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{dz}{z^{4}+1}$ along the circle $x^{2}+y^{2} = 2x$ , taking the path counterclockwise.

Problem 5428 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{dz}{(z-1)^{2}(z^{2}+1)}$ along the circle $x^{2}+y^{2} = 2x+2y$ , taking the path counterclockwise.

Problem 5429 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{dz}{(z^{2}-1)^{2}(z-3)^{2}}$ along the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}} = 2^{\frac{2}{3}}$ , taking the path counterclockwise.

Problem 5430 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int_{\left|z\right|=1} \frac{dz}{(z-a)^{n}(z-b)^{n}}$ ; $\left|a\right| < 1$ , $\left|b\right| < 1$ , taking the path counterclockwise.

Problem 5431 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{dz}{(z^{2}-1)(z^{3}-1)}$ along the circle $\left|z\right| = r \neq 1$ , taking the path counterclockwise.

Problem 5432 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{dz}{z^{3}(z^{10}-2)}$ along the circle $\left|z\right| = 2$ , taking the path counterclockwise.

Problem 5433 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Compute the integral $\displaystyle \int \frac{z^{2}}{e^{2\pi i z^{3}}-1}\, dz$ along the edge of the half-circle $\left|z\right| = R$ , under the conditions $n < R^{3} < n+1$ , where $n$ is an integer, taking the path counterclockwise.

Problem 5434 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

Find the integral $\displaystyle \int \frac{dt}{(z^{4}+1)\sqrt{z^{2}+1}}$ , taken along the parabola $x = y^{2}$ in the direction of increasing $y$ , taking the value of $\sqrt{z^{2}+1}$ to equal one at $z = 0$ .

Problem 5435 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.3 "Residues and Their Applications"

For what relation between $a$ and $b$ will the function

\int _{0}^{t} \frac{at\sin t+b\cos t}{t^{2}}\, dt

be single-valued?

Problem 14.1 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 14 "Computing Integrals Over a Closed Contour"

Compute the integrals of rational functions:

(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)

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

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.