Math Problems Vault

Problem 5.3.1

Evaluate the integral

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

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Problem 5.3.2

Evaluate

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

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

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Problem 5.3.3

For what value of $a$ is the function

f(z) = \int \left(\frac{1}{z} + \frac{a}{z^2}\right) \cos z\, dz

single-valued?

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Problem 5.3.4

Define

h(z) = \int _0^\infty \frac{\cos (t^2)}{e^t - zt}\, dt.

What is the largest possible $P$ so that $h(z)$ is analytic for $\left|z\right| < P$ ?

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Problem 5.3.5

Let $f(z)$ be analytic in $S = \left\{ z \in \mathbb {C}; \left|z\right| < 2\right\}$ . Show that

\frac{2}{\pi } \int _0^{\pi /2} f(e^{it}) \cos ^2\frac t2\, dt = 2f(0) + f'(0).

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Problem 5.3.6

Suppose that the real-valued function $u$ is harmonic in the disk $\left\{ \left|z\right| < 2\right\}$ , $v$ is its harmonic conjugate and $u(0) = v(0) = 0$ . Show that

\int _\gamma u^2(z) v^2(z) \frac{dz}{z} = \frac16 \int _\gamma (u^4(z) + v^4(z)) \frac{dz}{z},

where $\gamma (t) = e^{2\pi i t}$ , $t \in [0,1]$ .

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Problem 5.3.7

(a)(b)

Let $f$ be an analytic function on an open set containing $\overline{D(0,1)} = \left\{ z; \left|z\right| \leq 1\right\}$ .

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(a)

Prove that

\frac{d^n f}{dz^n}(0) = \frac{n!}{\pi } \int _0^{2\pi } e^{-ni\theta } [\mathrm{Re}\left(f(e^{i\theta })\right)]\, d\theta .

(b)

If $f(0) = 1$ , and if $\mathrm{Re}\left(f(z)\right) > 0$ for all points $z \in D(0,1)$ , prove that

\left|\frac{d^n f}{dz^n}(0)\right| \leq 2(n!).

Problem 5.3.8

If $f$ is analytic in the unit disk and its derivative satisfies

\left|f'(z)\right| \leq (1-\left|z\right|^2)^{-1},

show that the coefficients in the expansion

f(z) = \sum _{n=0}^{\infty } a_n z^n

satisfy $\left|a_n\right| \leq e$ for $n \geq 1$ , where $e$ is the base of natural logarithms.

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Problem 5.3.9

(a)(b)

Let $f = u+iv$ be an entire function.

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(a)

Show that if $u^2(z) \geq v^2(z)$ for all $z \in \mathbb {C}$ , then $f$ must be a constant.

(b)

Show that if $\left|f(z)\right| \leq A + B\left|z\right|^h$ for all $z \in \mathbb {C}$ with some positive numbers $A, B, h$ , then $f(z)$ is a polynomial of degree bounded by $h$ .

Problem 5.3.10

Let $f$ be an entire function that satisfies $\left|\mathrm{Re}\left(f(z)\right)\right| \leq \left|z\right|^n$ for all $z$ , where $n$ is a positive integer. Show that $f$ is a polynomial of degree at most $n$ .

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Problem 5.3.11

Compute the double integral

\iint _D \cos z\, dx\, dy

where $D$ is the disk given by $\left\{ z = x+iy \in \mathbb {C} : x^2+y^2 < 1\right\}$ .

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Problem 5.3.12

(a)(b)

Let

f(z) = \sum _{n=0}^{\infty } a_n z^n

be analytic in $D = \left\{ \left|z\right| < 1\right\}$ and assume that the integral

A = \iint _D \left|f'(z)\right|^2\, dx\, dy

is finite.

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(a)

Express $A$ in terms of the coefficients $a_n$ .

(b)

Prove that

\left|f(z) - f(0)\right| \leq \sqrt{\frac{A}{\pi } \log \frac{1}{1-\left|z\right|^2}}

for $z \in D$ .

Problem 5.3.13

Let $f$ be analytic in $\left\{ 0 < \left|z\right| < 1\right\}$ and in $L^2$ with respect to planar Lebesgue measure. Is $0$ a removable singularity? Proof or counterexample.

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Problem 5.3.14

Evaluate the integral

\int _0^{2\pi } \frac{d\theta }{(\rho - \cos \theta )^2 + \sin ^2\theta }, \qquad \rho \text{ a positive constant},

considering separately the cases $\rho < 1$ , $\rho > 1$ .

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Problem 5.3.15

Evaluate

\int _0^\pi \frac{dx}{a+\sin ^2 x}, \qquad a > 1,

by the method of residues.

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Problem 5.3.16

(a)(b)(c)

Consider the function

g(z,\theta ) = \frac{1}{1+z^2-2z\sin \theta }.

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(a)

Use the residue theorem to find an explicit formula for

f(z) = \int _0^{2\pi } g(z,\theta )\, d\theta

when $\left|z\right| < 1$ .

(b)

Integrate the Taylor expansion

g(z,\theta ) = \sum _{n=0}^{\infty } a_n(\theta ) z^n

term by term to find the coefficients in the Taylor expansion

f(z) = \sum _{n=0}^{\infty } f_n z^n.

(c)

Verify directly that (a) and (b) agree when $\left|z\right| < 1$ .

Problem 5.3.17

If $a$ is real, show that

\lim _{R\to \infty } \int _{-R}^{R} e^{-(x+ia)^2}\, dx

exists and is independent of $a$ .

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Problem 5.3.18

Let $n \geq 2$ be an integer. Compute

\int _0^\infty \frac{1}{1+x^n}\, dx.

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Problem 5.3.19

Evaluate

\int _0^\infty \cos (x^2)\, dx

with full justification.

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Problem 5.3.20

Evaluate

\int _0^\infty \frac{\sin ^2 x}{x^2}\, dx.

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Problem 5.3.21

Let $f(z)$ be holomorphic in the unit disk $\left|z\right| < 1$ . Prove that

\int _0^1 f(x)\, dx = \frac{1}{2\pi i} \int _{\left|z\right|=1} f(z) \log z\, dz,

where respective integration goes along the straight line from $0$ to $1$ and along the positively oriented unit circle starting from the point $z=1$ . The branch of $\log$ is chosen to be real for positive $z$ .

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Problem 5.3.22

Find the value of

\int _0^{2\pi } \log \left|a+be^{i\phi }\right|\, d\phi

where $a$ and $b$ are complex constants, not both equal to zero.

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Problem 5.3.23

Evaluate

\int _0^\infty \frac{\log x}{(1+x)^3}\, dx.

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Problem 5.3.24

(a)(b)

Evaluate the following integrals:

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(a)

$\displaystyle \int_{-i\infty }^{i\infty } \frac{dz}{(z^2-4)\log (z+1)}$ (the integration is over the imaginary axis),

(b)

$\displaystyle \int_0^\infty \frac{x^\alpha }{x^3+1}\, dx$ for $\alpha$ in the range $-1 < \alpha < 2$ .

Problem 5.3.25

Show that

\int _0^\infty \frac{x^\alpha }{(1+x^2)^2}\, dx = \frac{\pi (1-\alpha )}{4\cos \left(\frac{\pi \alpha }{2}\right)},

for $-1 < \alpha < 3$ , $\alpha \neq 1$ . What happens if $\alpha = 1$ ?

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Problem 5.3.26

(a)(b)

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(a)

Prove that

\int _0^\infty e^{ix} x^{-\alpha }\, dx

converges if $0 < \alpha < 1$ .

(b)

Use complex integration to show that

\int _0^\infty x^{-\alpha } \cos x\, dx = \sin \frac{\pi \alpha }{2} \cdot \Gamma (1-\alpha ).

Problem 5.3.27

Use a change of contour to show that

\int _0^\infty \frac{\cos (\alpha x)}{x+\beta }\, dx = \int _0^\infty \frac{e^{-\alpha x}(x-\beta )}{x^2+\beta ^2}\, dx,

provided that $\alpha$ and $\beta$ are positive. Define the left side as a limit of a proper integral and show that the limit exists.

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Problem 5.3.28

(a)(b)

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(a)

Let $c$ be the unit circle in the complex plane, and let $f$ be a continuous $\mathbb {C}$ -valued function on $c$ . Show that

F_f(z) = \frac{1}{2\pi i} \int _c \frac{f(\zeta )}{\zeta - z}\, d\zeta

is a holomorphic function of $z$ in the interior of the unit disk.

(b)

Find a continuous $f$ on $c$ which is not identically zero, but so that the associated function $F_f$ is identically zero.

Problem 5.3.29

Let $[a,b]$ be a finite interval in $\mathbb {R}$ and define, for $z$ in $D = \mathbb {C} - [a,b]$ ,

f(z) = \int _a^b \frac{dt}{t-z}.

Show that $f(z)$ is analytic in $D$ . Given $c$ , $a < c < b$ , calculate the limit of $f(z)$ as $z$ tends to $c$ from the upper half plane and as $z$ tends to $c$ from the lower half plane.

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Problem 5.3.30

For each $z \in U = \left\{ z : \im z > 0\right\}$ define

g(z) = \frac{1}{2\pi i} \int _{-1}^{1} \frac{\sin ^2 t}{t-z}\, dt.

Determine which points $a \in \mathbb {R}$ have the following property: there exist $\varepsilon > 0$ and an analytic function $f$ on $D(a,\varepsilon )$ such that $f(z) = g(z)$ for all $z \in U \cap D(a,\varepsilon )$ .

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Problem 5.3.31

Prove that there exists a constant $C$ such that for every polynomial $P$ ,

\int _{-\frac12}^{\frac12} P(x)\, dx \leq C \int _{\left|z\right|=1} \left|P(z)\right| \cdot \left|dz\right|.

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Problem 5.3.32

Let $p(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1 z + a_0$ be a complex polynomial. Show that there must be at least one point with $\left|z\right| = 1$ and $\left|p(z)\right| \geq 1$ .

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Problem 5.3.33

Let $f$ be analytic and absolutely integrable with respect to the Lebesgue area measure $dA = dx\, dy$ in the unit disk $\Delta$ . Prove that for $a \in \Delta$ ,

f'(a) = \frac{1}{\pi } \int _\Delta \frac{f(z)}{(1-\bar a z)^3}\, dA(z).

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Problem 5.3.34

If $f$ is analytic in the unit disk $\Delta$ , continuous in $\closure \Delta$ and maps $\Delta$ into itself, prove that for every point $a \in \Delta$ ,

\left|f'(a)\right| \leq \frac{6(1+\left|a\right|)}{(1-\left|a\right|)^3}.

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Problem 5.3.35

Let $f$ be holomorphic in a neighborhood of $\closure U$ , $U = \left\{ z : \left|z\right| < 1\right\}$ . Let $f(z) = \sum_{k=0}^{\infty } a_k z^k$ and $s_n(z) = \sum_{k=0}^{n} a_k z^k$ . Show that, among all polynomials $P$ of degree $n$ or less, the integral $\dfrac {1}{2\pi }\displaystyle \int_0^{2\pi } \left|f(e^{i\theta }) - P(e^{i\theta })\right|^2\, d\theta$ attains its minimum for $P = s_n$ .

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Problem 5.3.36

Suppose $f$ is analytic in the unit disk $D$ . For $z \in D$ and $0 < r < 1-\left|z\right|$ , let $B(z,r) = \left\{ w \in D : \left|w-z\right| < r\right\}$ . Show that $\left|f(z)\right|^n \leq \dfrac {1}{\pi r^2}\displaystyle \int_{B(z,r)} \left|f(w)\right|^n\, dA(w)$ , where $dA(w)$ is area measure on $D$ .

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Problem 5.3.37

Let $f$ be a smooth bounded function in the unit disk $\Delta$ and $F(z)$ be given by

F(z) = \frac1\pi \iint _\Delta \frac{f(w)}{w-z}\, dA(w).

Prove that $F(z)$ is analytic outside of $\Delta$ and for every $z \in \Delta$ ,

\frac{\partial F}{\partial \bar z} = f(z).

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