Math Problems Vault

Problem 5.10.1

Let $r_{1}, r_{2}, \ldots , r_{n}$ be distinct complex numbers. Show that a rational function of the form

f(z) = \frac{b_{0} + b_{1}z + \cdots + b_{n-2}z^{n-2} + b_{n-1}z^{n-1}}{(z-r_{1})(z-r_{2}) \cdots (z-r_{n})}

can be written as a sum

f(z) = \frac{A_{1}}{z-r_{1}} + \frac{A_{2}}{z-r_{2}} + \cdots + \frac{A_{n}}{z-r_{n}}

for suitable constants $A_{1}, \ldots , A_{n}$ .

?

Problem 5.10.2

Let

\cot (\pi z) = \sum _{n=-\infty }^{\infty } a_{n}z^{n}

be the Laurent expansion for $\cot (\pi z)$ on the annulus $1 < \left|z\right| < 2$ . Compute the $a_{n}$ 's.

?

Problem 5.10.3

Show that there is a complex analytic function defined on the set $U = \left\{ z \in \mathbb {C} \mid \left|z\right| > 4\right\}$ whose derivative is

\frac{z}{(z-1)(z-2)(z-3)} .

Is there a complex analytic function on $U$ whose derivative is

\frac{z^{2}}{(z-1)(z-2)(z-3)} \ ?

?

Problem 5.10.4

Let $n$ be a positive integer. Prove that the polynomial

f(x) = 1 + x + \frac{x^{2}}{2!} + \cdots + \frac{x^{m}}{m!}

in $\mathbb {R}^{x}$ has $n$ distinct complex zeros, $z_{1}, z_{2}, \ldots , z_{n}$ , and that they satisfy

\sum _{i=1}^{n} z_{i}^{k} = 0 \qquad \text{for } 0 \leq k \leq n-2 .

?

Problem 5.10.5

Let $P$ and $Q$ be complex polynomials with the degree of $Q$ at least two more than the degree of $P$ . Prove there is an $r > 0$ such that if $C$ is a closed curve outside $\left|z\right| = r$ , then

\int _{C} \frac{P(z)}{Q(z)} \, dz = 0 .

?

Problem 5.10.6

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 5.10.7

Evaluate the integral

I = \frac{1}{2\pi i} \int _{\left|z\right|=1} \frac{(z+2)^{2}}{z^{2}(2z-1)} \, dz

where the direction of integration is counterclockwise.

?

Problem 5.10.8

Let $a > 0$ be a constant $\neq 2$ . Let $C_{a}$ denote the positively oriented circle of radius $a$ centered at the origin. Evaluate

\int _{C_{a}} \frac{z^{2}+e^{z}}{z^{2}(z-2)} \, dz .

?

Problem 5.10.9

Let $r$ be a positive number and $a$ a complex number such that $\left|a\right| \neq r$ . Evaluate the integral

I(a,r) = \int _{\left|z\right|=r} \frac{\overline{z}}{a-z} \, dz

where the circle $\left|z\right| = r$ has the counterclockwise orientation.

?

Problem 5.10.10

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 5.10.11

Evaluate the integral

I = \frac{1}{2\pi i} \int _{\left|z\right|=1} \frac{1}{\sin 4z} \, dz

where the direction of integration is counterclockwise.

?

Problem 5.10.12

Evaluate the integrals

I_{n} = \frac{1}{2\pi i} \int _{C_{n}} \frac{1}{z^{3} \sin z} \, dz, \qquad n = 0, 1, \ldots ,

where $C_{n}$ is the circle $\left|z\right| = (n+\frac{1}{2})\pi$ with the counterclockwise orientation.

?

Problem 5.10.13

Compute

\frac{1}{2\pi i} \int _{C} \frac{dz}{\sin \frac{1}{z}}

where $C$ is the circle $\left|z\right| = \frac{1}{5}$ , positively oriented.

?

Problem 5.10.14

(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 5.10.15

Let $C$ be a simple closed contour enclosing the points $0, 1, 2, \ldots , k$ in the complex plane, with positive orientation. Evaluate the integrals

I_{k} = \int _{C} \frac{dz}{z(z-1) \cdots (z-k)}, \qquad k = 0, 1, \ldots ,

J_{k} = \int _{C} \frac{(z-1) \cdots (z-k)}{z} \, dz, \qquad k = 0, 1, \ldots .

?

Problem 5.10.16

Evaluate

\int _{\left|z\right|=1} (e^{2\pi z}+1)^{-2} \, dz

where the integral is taken in counterclockwise direction.

?

Problem 5.10.17

Evaluate

\frac{1}{2\pi i} \int _{\left|z\right|=1} \frac{z^{11}}{12z^{12} - 4z^{9} + 2z^{6} - 4z^{3} + 1} \, dz

where the direction of integration is counterclockwise.

?

Problem 5.10.18

Evaluate

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

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

?

Problem 5.10.19

Evaluate the integral

I = \frac{1}{2\pi i} \int _{C} \frac{dz}{(z-2)(1+2z)^{2}(1-3z)^{3}}

where $C$ is the circle $\left|z\right| = 1$ with counterclockwise orientation.

?

Problem 5.10.20

Evaluate

I = \frac{1}{2\pi i} \int _{C} \frac{1}{3z^{n}-1} \, dz ,

where $n$ is a positive integer and $C$ is the circle $\left|z\right| = 1$ , with counterclockwise orientation.

?

Problem 5.10.21

Evaluate

\int _{C} \frac{e^{z}}{z(2z+1)^{2}} \, dz ,

where $C$ is the unit circle with counterclockwise orientation.

?

Problem 5.10.22

Let $a = \dfrac {1+i}{2}$ , $b = \dfrac {-1+i}{2}$ . Let $\Gamma$ be the polygonal path in the complex plane with successive vertices $-1, -1+i, 1, 1+i, -1$ . Evaluate the integral

I = \int _{\Gamma } \frac{1}{(z-a)^{2}(z-b)^{3}} \, dz .

?

Problem 5.10.23

Evaluate the integral $\dfrac {1}{2\pi i} \int_{\gamma } f(z) \, dz$ for the function $f(z) = z^{-2}(1-z^{2})^{-1}e^{z}$ and the curve $\gamma$ depicted by {#fig-1 width="60%"} a curve that starts as a large loop enclosing $-1$ and $0$ , crosses itself, and continues as a small loop enclosing only $1$ , so that its total winding numbers about $-1$ , $0$ , $1$ are as read off the picture.

?

Problem 5.10.24

Evaluate

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

where $C$ is the closed curve shown below: {#fig-1 width="60%"} a figure-of-eight curve that winds twice, negatively, around a small loop enclosing $0$ , and twice, positively, around a loop enclosing $1$ .

?

Problem 5.10.25

Let $n$ be a positive integer and $0 < \theta < \pi$ . Prove that

\frac{1}{2\pi i} \int _{\left|z\right|=2} \frac{z^{n}}{1-2z\cos \theta +z^{2}} \, dz = \frac{\sin n\theta }{\sin \theta }

where the circle $\left|z\right| = 2$ is oriented counterclockwise.

?

Problem 5.10.26

Use the Residue Theorem to evaluate the integral

I(a) = \int _{0}^{2\pi } \frac{d\theta }{a+\cos \theta }

where $a$ is real and $a > 1$ . Why the formula obtained for $I(a)$ is also valid for certain complex (nonreal) values of $a$ ?

?

Problem 5.10.27

Evaluate

\int _{0}^{2\pi } \frac{d\theta }{1-2r\cos \theta +r^{2}}

where $r^{2} \neq 1$ .

?

Problem 5.10.28

Evaluate

\int _{0}^{\pi } \frac{\cos 4\theta }{1+\cos ^{2}\theta } \, d\theta .

?

Problem 5.10.29

Evaluate the integral

\int _{0}^{2\pi } \frac{\cos ^{2}3\theta }{5-4\cos 2\theta } \, d\theta

?

Problem 5.10.30

Evaluate

I = \int _{\left|z\right|=1} \frac{\cos ^{3}z}{z^{3}} \, dz

where the direction of integration is counterclockwise.

?

Problem 5.10.31

Let $n$ be a positive integer. Compute

\int _{0}^{2\pi } \frac{1-\cos n\theta }{1-\cos \theta } \, d\theta .

?

Problem 5.10.32

Evaluate the integrals

\int _{-\pi }^{\pi } \frac{\sin n\theta }{\sin \theta } \, d\theta , \qquad n = 1, 2, \ldots .

?

Problem 5.10.33

For $a > 1$ and $n = 0, 1, 2, \ldots$ , evaluate the integrals

C_{n}(a) = \int _{-\pi }^{\pi } \frac{\cos n\theta }{a-\cos \theta } \, d\theta , \qquad S_{n}(a) = \int _{-\pi }^{\pi } \frac{\sin n\theta }{a-\cos \theta } \, d\theta .

?