Math Problems Vault

Problem 8.1

Prove that the sums of the following series are regular in the domains indicated in parentheses:

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(1)

$\sum_{n = 0}^{\infty } \frac{\cos n z}{n!} \quad (\left|z\right| < \infty )$ ;

(2)

$\sum_{n = 1}^{\infty } \frac{n!}{n^{n}} \sin n z(\left|\mathrm{Im}\left(z\right)\right| < 1)$ ;

(3)

$\sum_{n = 1}^{\infty }\left[\frac{z(z+n)}{n}\right]^{n} \quad (\left|z\right| < 1)$ ;

(4)

$\sum_{n = 1}^{\infty } \frac{1}{n(z-n)}(z \neq 1,2, \ldots )$ ;

(5)

$\sum_{n = 1}^{\infty } \frac{(-1)^{n}}{z+n}(z \neq -1,-2, \ldots )$ ;

(6)

$\sum_{n = 1}^{\infty }(-1)^{n} n^{-z} \quad (\mathrm{Re}\left(z\right) > 0)$ ;

(7)

$\sum_{n = -\infty }^{\infty } 2^{-n^{2} z} n^{n} \quad (\mathrm{Re}\left(z\right) > 0)$ ;

(8)

$\sum_{n = 0}^{\infty } e^{-z^{2} \sqrt{n}} \quad (\left|\arg z\right| < \pi / 4)$ .

Problem 8.2

Let $\left\{ \lambda_{n}\right\}$ be an increasing sequence of positive numbers, and let

\varlimsup _{n \rightarrow \infty } \frac{n}{\lambda _{n}} = \alpha > 0, \quad \text{ and } \quad \varlimsup _{n \rightarrow \infty }\left|a_{n}\right|^{1 / n} = \rho > 0

Prove that the sum of the series

\sum _{n = 1}^{\infty } a_{n} e^{-\lambda _{n} z}

is regular in the half-plane $\mathrm{Re}\left(z\right) > \alpha \ln \rho$ .

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Problem 8.3

Prove that the following infinite products represent functions regular in the domains indicated in parentheses:

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(1)

$\prod_{n = 1}^{\infty }\left(1-z^{n}\right) \quad (\left|z\right| < 1)$ ;

(2)

$\prod_{n = 1}^{\infty }\left(1+z^{2 n} n^{z}\right) \quad (\left|z\right| < 1)$ ;

(3)

$\prod_{n = 1}^{\infty }\left(1-\frac{z}{n}\right) e^{z / n} \quad (\left|z\right| < \infty )$ ;

(4)

$\prod_{n = 1}^{\infty }\left(1+(-1)^{n} \cdot \frac{z}{n}\right) \quad (\left|z\right| < \infty )$ ;

(5)

$\prod_{n = 1}^{\infty } \cos \frac{z}{n} \quad (\left|z\right| < \infty )$ ;

(6)

$\prod_{n = 1}^{\infty } \operatorname {th} \frac{n^{2}}{z^{2}} \quad \left(\left|\arg z\right| < \frac{\pi }{4}\right)$ .

Problem 8.4

Let the series $\sum_{k = 1}^{\infty }\left|c_{k}\right|$ converge, and let the function $u(z)$ be regular in a domain $D$ . Prove that the infinite product

\prod _{k = 1}^{\infty }\left(1+c_{k} u(z)\right)

represents a function regular in the domain $D$ .

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Problem 8.5

Let the series converge

\sum _{k = 1}^{\infty } c_{k}, \sum _{k = 1}^{\infty } c_{k}^{2}, \ldots , \sum _{k = 1}^{\infty } c_{k}^{m}, \sum _{k = 1}^{\infty }\left|c_{k}\right|^{m+1}.

Prove that the infinite product

\prod _{n = 1}^{\infty }\left(1+c_{n} n^{-z}\right)

represents a function regular in the half-plane $\mathrm{Re}\left(z\right) > 0$ .

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Problem 8.6

Prove the regularity of the functions represented by the following integrals in the domains indicated in parentheses:

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(1)

$\int_{0}^{\infty } \frac{e^{-i t}}{1+t^{2}} d t \quad (\mathrm{Re}\left(z\right) > 0)$ ;

(2)

$\int_{1}^{\infty } t^{z-1} e^{-t} d t \quad (\left|z\right| < \infty )$ ;

(3)

$\int_{0}^{\infty } t^{z-1} e^{-t} d t \quad (\mathrm{Re}\left(z\right) > 0)$ ;

(4)

$\int_{1}^{\infty } \frac{t^{z-1}}{t^{2}+1} d t \quad (\mathrm{Re}\left(z\right) < 2)$ ;

(5)

$\int_{0}^{\infty } \frac{t^{z-1}}{t^{2}+1} d t \quad (0 < \mathrm{Re}\left(z\right) < 2)$ ;

(6)

$\int_{0}^{\infty } \frac{d t}{\left(t^{2}+1\right)\left(1+e^{t z}\right)} \quad (\mathrm{Re}\left(z\right) > 0)$ ;

(7)

$\int_{0}^{\infty } \frac{e^{-t} d t}{e^{2 \pi i t z}+1} \quad (\mathrm{Im}\left(z\right) > 0)$ ;

(8)

$\int_{0}^{1} t^{z}(1-t)^{1-z} d t \quad (-1 < \mathrm{Re}\left(z\right) < 2)$ ;

(9)

$\int_{0}^{1} \frac{\cos t z}{z+t} d t \quad (z \notin [-1,0], z \neq \infty )$ ;

(10)

$\int_{0}^{\infty } \frac{t \sin t}{t^{2}+z^{2}} \quad (\mathrm{Re}\left(z\right) > 0)$ .

Problem 8.7

Let the function $\varphi (t)$ be continuous for $t \geqslant 0$ and satisfy the condition

\varlimsup _{t \rightarrow +\infty } \frac{\ln \left|\varphi (t)\right|}{t} = \sigma .

Prove that the integral

\int _{0}^{\infty } \varphi (t) e^{-t z} d t

represents a function regular in the half-plane $\mathrm{Re}\left(z\right) > \sigma$ .

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Problem 8.8

Let the function $\varphi (t)$ be continuous for $-\infty < t < \infty$ and satisfy the conditions

\varlimsup _{t \rightarrow +\infty } \frac{\ln \left|\varphi (t)\right|}{t} = \sigma _{1}, \quad \varlimsup _{t \rightarrow -\infty } \frac{\ln \left|\varphi (t)\right|}{t} = \sigma _{2}, \quad \sigma _{2} > \sigma _{1}.

Prove that the integral

\int _{-\infty }^{\infty } \varphi (t) e^{i t z} d t

represents a function regular in the strip $\sigma_{1} < \mathrm{Im}\left(z\right) < \sigma_{2}$ .

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Problem 8.9

Find the domains of regularity of the functions represented by the following integrals:

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(1)

$\int_{0}^{\infty } \frac{\varphi (t)}{t+z} d t,\left|\varphi (t)\right| \leqslant M(1+t)^{-\alpha }, \alpha > 0$ ;

(2)

$\int_{1}^{\infty } \frac{\varphi (t)}{t^{2}-z^{2}} d t,\left|\varphi (t)\right| \leqslant M$ ;

(3)

$\int_{0}^{\infty } \frac{\varphi (t)}{e^{t}+z} d t, \quad \left|\varphi (t)\right| \leqslant M(1+t)^{m}, \quad m < \infty$ ;

(4)

$\int_{-\infty }^{\infty } \frac{\varphi (t)}{e^{t}+z} d t,\left|\varphi (t)\right| \leqslant \frac{M}{1+t^{2}}$ ;

(5)

$\int_{1}^{\infty } \frac{\varphi (t)}{\left(t^{3}-z^{3}\right)^{2}} d t,\left|\varphi (t)\right| \leqslant M t^{\alpha }, \alpha < 5$ ;

(6)

$\int_{0}^{1} \frac{\varphi (t)}{t-z^{3}} d t,\left|\varphi (t)\right| \leqslant M t^{\alpha }, \alpha > -1$ ;

(7)

$\int_{\gamma } \frac{e^{-t}}{t+z} d t \quad (\gamma = \left\{ t: \arg t = \alpha \right\} ), \quad -\pi < \alpha < \pi$ ;

(8)

$\int_{\gamma } \frac{t^{2} d t}{e^{t}+z} \quad (\gamma = \left\{ t: \arg t = \alpha \right\} ), 0 \leqslant \alpha \leqslant \frac{\pi }{2}$ ;

(9)

$\int_{\gamma } \frac{\cos t}{e^{t}-z}\left(\gamma = \left\{ t: \arg t = \frac{\pi }{2}, \quad 0 \leqslant \left|t\right| \leqslant \pi \right\} \right)$ .

Problem 8.10

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$ , be regular in the disk $\left|z\right| < 1$ and satisfy the conditions

\left|f_{n}(z)\right| \leqslant M(1-\left|z\right|)^{-m} \quad (\left|z\right| < 1, n = 1,2, \ldots ),

where the numbers $M$ and $m$ do not depend on $n$ . Prove that from the sequence $\left\{ f_{n}(z)\right\}$ one can select a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the disk $\left|z\right| < 1$ .

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Problem 8.11

Let the boundary $\partial D$ of a bounded domain $D$ consist of a finite number of simple piecewise-smooth curves, and let functions $\varphi_{n}(\zeta )$ be defined and continuous on $\partial D$ . Prove that if the sequence $\left\{ \varphi_{n}(\zeta )\right\}$ is uniformly bounded on $\partial D$ , then from the sequence of functions $\left\{ f_{n}(z)\right\}$ ,

f_{n}(z) = \frac{1}{2 \pi i} \int _{\partial D} \frac{\varphi _{n}(\zeta )}{(\zeta -z)^{m}} d \zeta

( $m$ a fixed integer) one can extract a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the domain $D$ .

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Problem 8.12

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$ , be regular in a domain $D$ and satisfy the conditions $\mathrm{Re}\left(f_{n}(z)\right) \geqslant 0 \quad (z \in D, n = 1,2, \ldots )$ . Prove that from the sequence $\left\{ f_{n}(z)\right\}$ one can extract a subsequence $\left\{ f_{n_{k}}(z)\right\}$ converging uniformly on every compact subset of the domain $D$ (possibly to infinity).

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Problem 8.13

Let the functions $f_{n}(z), n = 1,2, \ldots$ , be regular and uniformly bounded in a domain $D$ . Prove that if the sequence $\left\{ f_{n}(z)\right\}$ converges on a set $E$ having at least one limit point in the domain $D$ , then it converges uniformly on every compact subset of the domain $D$ . (Vitali's theorem.)

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Problem 8.14

Let the functions $u_{n}(x, y), n = 1,2, \ldots$ , be harmonic in a domain $D$ , and let the sequence $\left\{ u_{n}(x, y)\right\}$ converge uniformly in the domain $D$ to a function $u(x, y)$ . Prove that the function $u(x, y)$ is harmonic in the domain $D$ .

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Problem 8.15

Let the functions $u_{n}(x, y), n = 1,2, \ldots$ , be harmonic in a domain $D$ and satisfy the conditions

\left|u_{n}(x, y)\right| \leqslant M \quad ((x, y) \in D, n = 1,2, \ldots ).

Prove that from the sequence $\left\{ u_{n}(x, y)\right\}$ one can extract a subsequence $\left\{ u_{n_{k}}(x, y)\right\}$ converging uniformly on every compact subset of the domain $D$ .

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Problem 8.16

Let the functions $u_{n}(x, y), n = 1,2, \ldots$ , be harmonic in a domain $D$ and satisfy the conditions

u_{n+1}(x, y) \geqslant u_{n}(x, y) \quad ((x, y) \in D, n = 1,2, \ldots )

Prove that the sequence $\left\{ u_{n}(x, y)\right\}$ converges uniformly on every compact subset of the domain $D$ (possibly to $+\infty$ ).

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Problem 8.17

Let the functions $f_{n}(z), \quad n = 1,2, \ldots$ , be regular in a domain $D$ , not vanish in this domain, and satisfy the inequalities

\left|f_{n}(z)\right| \leqslant M^{n} \quad (z \in D, n = 1,2, \ldots )

(the constant $M$ does not depend on $n$ ). Prove that from the sequence $\left\{ \sqrt[n]{\left|f_{n}(z)\right|}\right\}$ one can extract a subsequence converging uniformly on every compact subset of the domain $D$ to a function $\left|g(z)\right|$ , where $g(z)$ is a function regular in the domain $D$ .

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Problem 8.18

Let the power series $\sum_{n = 0}^{+\infty } c_{n} z^{n}$ have radius of convergence $R, 0 < R < \infty$ . Prove that for each point

z_{0} = R e^{i \varphi }

there exist sequences $\left\{ n_{k}\right\}$ and $\left\{ z_{k}\right\}$ for which

\lim _{k \rightarrow \infty } z_{k} = z_{0} ; \quad \sum _{n = 0}^{n_{k}} c_{n} z_{k}^{n} = 0

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