Math Problems Vault

Problem 9.1

Let the function $f(z)$ be regular in the closure $\bar{G}$ of a domain $G$ , and $f(z) \not\equiv$ const. Prove that only a finite number of solutions of the equation $f(z) = a$ (for an arbitrary fixed value $a$ ) lie in the domain $G$ .

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Problem 9.2

Does there exist a function $f(z)$ , regular in some neighborhood of the point $z = 0$ and satisfying one of the following conditions (for all $n = 1,2, \ldots )$ :

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(1)

$f\left(\frac{1}{n}\right) = \sin \frac{\pi n}{2}$ ;

(2)

$f\left(\frac{1}{n}\right) = \frac{1}{n} \cos \pi n$ ;

(3)

$f\left(\frac{1}{n}\right) = \frac{1}{2 n+1}$ ;

(4)

$f\left(\frac{1}{n}\right) = \frac{\cos^{2} \pi n}{2 n+1}$ ;

(5)

$f\left(\frac{1}{n}\right) = \frac{1}{2 n+\cos \pi n}$ ;

(6)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{1}{n^{2}}$ ;

(7)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{1}{2 n+1}$ ;

(8)

$f\left(\frac{1}{n}\right) = e^{-n}$ ;

(9)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{1}{n^{3}}$ ;

(10)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{\sin \frac{\pi n}{2}}{n^{3}}$ ;

(11)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{\cos^{2}(\pi n)}{n^{2}}$ ;

(12)

$f\left(\frac{1}{n}\right) = f\left(-\frac{1}{n}\right) = \frac{\sin^{2} \frac{\pi n}{2}}{n^{2}}$ ;

(13)

$\left|f\left(\frac{1}{n}\right)\right| < e^{-n}$ ;

(14)

$2^{-n} < \left|f\left(\frac{1}{n}\right)\right| < 2^{1-n}$ ;

(15)

$n^{-5 / 2} < \left|f\left(\frac{1}{n}\right)\right| < 2 n^{-5 / 2}$ ;

(16)

$\left|f\left(\frac{1}{n}\right)-\frac{\cos \pi n}{2 n+1}\right| < \frac{1}{n^{2}}$ .

Problem 9.3

Let the functions $f_{1}(z)$ and $f_{2}(z)$ be regular in a domain $D$ and satisfy in this domain the differential equation $f^{\prime }(z) = P(z, f(z))$ , where $P(z, w)$ is a polynomial in its variables. Prove that if at some point $z_{0} \in D$ the equality $f_{1}\left(z_{0}\right) = f_{2}\left(z_{0}\right)$ holds, then $f_{1}(z) \equiv f_{2}(z)$ .

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Problem 9.4

Let the functions $f_{1}(z)$ and $f_{2}(z)$ be regular in a domain $D$ and satisfy in this domain the differential equation

f^{(m)}(z) = P\left(z, f, f^{\prime }, \ldots , f^{(m-1)}\right),

where $P$ is a polynomial in its variables. Prove that if at some point $z_{0} \in D$ the equalities

f_{1}\left(z_{0}\right) = f_{2}\left(z_{0}\right), \ldots , f_{1}^{(m-1)}\left(z_{0}\right) = f_{2}^{(m-1)}\left(z_{0}\right)

hold, then $f_{1}(z) \equiv f_{2}(z)$ .

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Problem 9.5

Prove that the functional equation $f(z) = f(2 z)$ has no solutions regular at the point $z = 0$ and not identically constant.

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Problem 9.6

Let $q = e^{2 \pi i \alpha }$ , where $\alpha$ is an irrational number. Prove that the functional equation $f(z) = f(q z)$ has no solutions $f(z)$ regular for $1 / 2 < \left|z\right| < 2$ and not identically constant.

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Problem 9.7

Let $f(z)$ be a periodic function, regular in some domain containing the point $z = \infty$ . Prove that $f(z) =$ const throughout the domain of regularity.

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Problem 9.8

Prove that the functions $e^{z}, \cos z, \sin z, \operatorname {ch} z, \operatorname {sh} z$ , originally defined only for real values of the variable, can be regularly continued to the entire complex plane by expanding these functions in a Taylor series.

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Problem 9.9

Find the widest possible domain into which the following functions can be regularly continued from the real axis:

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(1)

$\operatorname {tg} z$ ;

(2)

$\operatorname {ctg} z$ ;

(3)

$\operatorname {th} z$ ;

(4)

$e^{1 / \cos z}$ ;

(5)

$e^{-\operatorname {tg} z}$ ;

(6)

$\sin (\operatorname {th} z)$ ;

(7)

$\cos \left(e^{z^{2}}\right)$ ;

(8)

$\operatorname {th}\left(e^{z}\right)$ ;

(9)

$\operatorname {ctg}(\operatorname {ch} z)$ .

Problem 9.10

Prove that the function $\ln z$ , defined for real positive values of $z$ , can be regularly continued to the entire complex plane with a cut along the negative part of the real axis, and, denoting this regular continuation by the symbol $h(z)$ , obtain for it the formula

h(z) = \ln \left|z\right|+i \arg z \quad (\left|\arg z\right| < \pi ).

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Problem 9.11

Let $\alpha$ be an arbitrary real number. Prove that the function $z^{\alpha }$ , defined for real positive $z$ , can be regularly continued to the entire complex plane with a cut along the negative part of the real axis, and, denoting this regular continuation by the symbol $\left(z^{\alpha }\right)$ , obtain for it the formula

\left(z^{\alpha }\right) = \left|z\right|^{\alpha } e^{i \alpha \arg z} \quad (\left|\arg z\right| < \pi ).

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Problem 9.12

Let the function $\Phi \left(z_{1}, \ldots , z_{n}\right)$ be defined when

z_{k} \in D_{k} \quad (k = 1,2, \ldots , n)

and be regular in each variable $z_{k}$ in the domain $D_{k}$ for arbitrary fixed values of the remaining variables. Prove that if each domain $D_{k}$ contains a nonempty interval $\left(a_{k}, b_{k}\right)$ of the real axis, and if

\Phi \left(z_{1}, \ldots , z_{n}\right) = 0 \quad \left(z_{1} \in \left(a_{1}, b_{1}\right), \ldots , z_{n} \in \left(a_{n}, b_{n}\right)\right),

then $\Phi \left(z_{1}, \ldots , z_{n}\right) \equiv 0$ .

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Problem 9.13

Relying on the validity of the formulas given below for real values of the variables, prove their validity also for arbitrary complex values of these variables:

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(1)

$e^{z_{1}+z_{2}} = e^{z_{1}} \cdot e^{z_{2}}$ ;

(2)

$\cos^{2} z+\sin^{2} z = 1$ ;

(3)

$\sin 2 z = 2 \sin z \cos z$ ;

(4)

$\operatorname {ch} 2 z = \operatorname {ch}^{2} z+\operatorname {sh}^{2} z$ ;

(5)

$\sin \left(z_{1}+z_{2}\right) = \sin z_{1} \cos z_{2}+\cos z_{1} \sin z_{2}$ ;

(6)

$\operatorname {ch}\left(z_{1}+z_{2}\right) = \operatorname {ch} z_{1} \operatorname {ch} z_{2}+\operatorname {sh} z_{1} \operatorname {sh} z_{2}$ ;

(7)

$\cos z_{1}+\cos z_{2} = 2 \cos \frac{z_{1}+z_{2}}{2} \cos \frac{z_{1}-z_{2}}{2}$ ;

(8)

$\operatorname {sh} z_{1}+\operatorname {sh} z_{2} = 2 \operatorname {sh} \frac{z_{1}+z_{2}}{2} \operatorname {ch} \frac{z_{1}-z_{2}}{2}$ .

Problem 9.14

Let the function $f(z)$ be regular in some domain containing the segment $[0,1]$ and satisfy the condition $f(z+1) = f(z)$ . Prove that the function $f(z)$ can be regularly continued into some strip $-\delta < \mathrm{Im}\left(z\right) < \delta$ , $\delta > 0$ .

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Problem 9.15

Let the function $f(z)$ be regular in some domain containing the segment $[0,1]$ and satisfy the condition

f(z+1) = z f(z)+p(z),

where $p(z)$ is a polynomial. Prove that the function $f(z)$ can be regularly continued into some strip $-\delta < \mathrm{Im}\left(z\right) < \delta , \delta > 0$ .

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Problem 9.16

Let the function $f(z)$ be regular in some domain containing the segment $[1,2]$ and satisfy the condition

f(2 z) = f(z)+p(z),

where $p(z)$ is a polynomial. Prove that the function $f(z)$ can be regularly continued into some angle $-\delta < \arg z < \delta , \delta > 0$ .

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Problem 9.17

Let the function $f(z)$ be regular in the annulus $\rho < \left|z\right| < 1, \rho > 0$ , and satisfy in it the functional equation

f(z) = f\left(z^{2}\right)+g(z),

where $g(z)$ is some given function, regular in the disk $\left|z\right| < 1$ . Prove that the function $f(z)$ can be regularly continued into the annulus $0 < \left|z\right| < 1$ .

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Problem 9.18

Euler's gamma function $\Gamma (z)$ is defined for real $z > 0$ by the equality

\Gamma (z) = \int _{0}^{\infty } t^{z-1} e^{-t} d t

Prove that the function $\Gamma (z)$ can be regularly continued to the entire complex plane, with the exception of the points $z = 0,-1,-2, \ldots$ .

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Problem 9.19

Euler's beta function $\mathrm{B}(z, \zeta )$ is defined for real $z > 0$ and $\zeta > 0$ by the equality

\mathrm{B}(z, \zeta ) = \int _{0}^{1} t^{z-1}(1-t)^{\zeta -1} d t

Prove that the function $\mathrm{B}(z, \zeta )$ can be regularly continued in each of the variables to the entire complex plane, with the exception of the points $z = 0,-1,-2, \ldots ($ respectively $\zeta = 0,-1,-2, \ldots )$ .

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Problem 9.20

Let the function $\varphi (\zeta )$ be regular in the annulus $r \leqslant \left|\zeta \right| \leqslant R$ . Prove that the function $f(z)$ , given in the disk $\left|z\right| < r$ by the formula

f(z) = \frac{1}{2 \pi i} \oint _{\left|\zeta \right| = r}(\zeta -z)^{m} \varphi (\zeta ) d \zeta \quad (\left|z\right| < r)

( $m$ an integer) can be regularly continued into the disk $\left|z\right| < R$ , and that this regular continuation $g(z)$ is given by the formula

g(z) = \frac{1}{2 \pi i} \oint _{\left|\zeta \right| = R}(\zeta -z)^{m} \varphi (\zeta ) d \zeta \quad (\left|z\right| < R)

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Problem 9.21

Prove that the following functions can be regularly continued to the domains $G$ indicated in parentheses:

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(1)

$f(z) = \oint_{\left|\zeta \right| = 1} e^{\left(\zeta +\frac{1}{\zeta }\right)} \frac{d \zeta }{\zeta -z},\left|z\right| < 1 \quad (G = \left\{ z:\left|z\right| < \infty \right\} ) ;$

(2)

$f(z) = \oint_{\left|\zeta \right| = 1} e^{\left(\zeta +\frac{1}{\zeta }\right)} \frac{d \zeta }{\zeta -z},\left|z\right| > 1 \quad (G = \left\{ z:\left|z\right| > 0\right\} )$ ;

(3)

$f(z) = \oint_{\left|\zeta \right| = 2} \frac{\cos \left(\zeta +\frac{1}{\zeta }\right)}{\zeta^{2}+1} \frac{d \zeta }{(\zeta -z)^{2}}, \quad \left|z\right| < 2 \quad (G = \left\{ z:\left|z\right| < \infty \right\} )$ ;

(4)

$f(z) = \oint_{\left|\zeta \right| = 2} \frac{\cos \left(\zeta +\frac{1}{\zeta }\right)}{\zeta^{2}+1} \frac{d \zeta }{(\zeta -z)^{2}},\left|z\right| > 2 \quad (G = \left\{ z: z \neq 0, i,-i\right\} )$ ;

(5)

$f(z) = \oint_{\left|\zeta \right| = 1} \operatorname {sh}\left(\zeta +\frac{1}{\zeta }\right) \frac{d \zeta }{\zeta^{2}+z^{2}},\left|z\right| < 1 \quad (G = \left\{ z:\left|z\right| < \infty \right\} )$ ;

(6)

$f(z) = \oint_{\left|\zeta \right| = 1} \frac{e^{-1 / \zeta }}{\operatorname {ch} \zeta } \frac{d \zeta }{\zeta^{2}+z^{2}},\left|z\right| < 1 \quad (G = \left\{ z: z \neq \pm \pi i / 2, \pm 3 \pi i / 2, \ldots \right\} )$ .

Problem 9.22

Let the function $\varphi (\zeta )$ be regular in the strip $-a \leqslant \mathrm{Re}\left(\zeta \right) \leqslant 0$ and satisfy the condition

\left|\varphi (\zeta )\right| \leqslant M(1+\left|\zeta \right|)^{-\alpha }, \quad \alpha > 0 \quad (-a \leqslant \mathrm{Re}\left(\zeta \right) \leqslant 0)

Prove that the function

f(z) = \int _{-i \infty }^{+i \infty } \frac{\varphi (\zeta )}{\zeta -z} d \zeta \quad (\mathrm{Re}\left(z\right) > 0)

admits a regular continuation into the half-plane $\mathrm{Re}\left(z\right) > -a$ , and that this continuation $g(z)$ is given by the formula

g(z) = \int _{-a-i \infty }^{-a+i \infty } \frac{\varphi (\zeta )}{\zeta -z} d \zeta \quad (\mathrm{Re}\left(z\right) > -a)

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Problem 9.23

Prove that for the regular continuation $g(z)$ , constructed in Problem 22, the formula also holds

g(z) = \int _{-i \infty }^{i \infty } \frac{\varphi (\zeta )}{\zeta -z} d \zeta -2 \pi i \varphi (z) \quad (-a \leqslant \mathrm{Re}\left(z\right) < 0)

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