Math Problems Vault

Problem 20.1

Prove that a function regular in a domain $G$ is a function analytic in the domain $G$ .

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Problem 20.2

Prove that if a function analytic in a domain $G$ does not depend on the shape of the piecewise-smooth oriented curve along which the analytic continuation is carried out, but depends only on its endpoint, then this function is regular in the domain $G$ .

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Problem 20.3

Let $F_{1}$ and $F_{2}$ be two analytic functions in a domain $G$ . Prove that the functions $F_{1}+F_{2}$ and $F_{1} \cdot F_{2}$ , defined by elements of the form $(B_{r}(a), f_{1}+f_{2})$ , $\left(B_{r}(a), f_{1} \cdot f_{2}\right)$ , where $\left(B_{r}(a), f_{1}\right)$ and $\left(B_{r}(a), f_{2}\right)$ are any elements of $F_{1}$ and $F_{2}$ , are also analytic functions in $G$ .

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Problem 20.4

Let $F_{1}$ and $F_{2}$ be two analytic functions in a domain $G$ , defined by elements of the form $(B_{r}(a), f_{1})$ and $(B_{r}(a), f_{2})$ , where for every element $\left(B_{r}(a), f_{2}\right)$ of the function $F_{2}$ the regular function $f_{2}(z) \neq 0, z \in B_{r}(a)$ . Prove that the function $\frac{F_{1}}{F_{2}}$ , defined by elements of the form $\left(B_{r}(a), \frac{f_{1}}{f_{2}}\right)$ , is analytic in the domain $G$ .

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Problem 20.5

Let $F$ be an analytic function in a domain $G$ , defined by elements of the form $(B_{r}(a), f)$ . Let $g: \mathbb {C} \rightarrow \mathbb {C}$ be a regular function on the plane $\mathbb {C}$ . Prove that the function defined by elements of the form $\left(B_{r}(a), g(f)\right)$ is analytic in the domain $G$ .

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Problem 20.6

For every point $a \in \mathbb {C} \backslash \left\{ 0\right\}$ and every piecewise-smooth oriented curve $\gamma_{a}$ starting at the point 1 and ending at the point $a$ , such that $0 \notin \gamma_{a}$ , let us define the quantity

h_{\gamma a}(a) = \int _{\gamma a} \frac{d \zeta }{\zeta }

and the function $h_{\gamma a}: B_{\left|a\right|}(a) \rightarrow \mathbb {C}$ of the form

h_{\gamma a}(z) = h_{\gamma a}(a)+\int _{a}^{z} \frac{d \zeta }{\zeta }, \quad z \in B_{\left|a\right|}(a)

Prove that the family of elements

\left(B_{\left|a\right|}(a), h_{\gamma a}\right), \quad a \in \mathbb {C} \backslash \left\{ 0\right\}

forms a function (denoted by $\operatorname {Ln} z$ ) analytic in the entire complex plane with the point $z = 0$ deleted.

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Problem 20.7

Let $\alpha$ be an arbitrary complex number. Prove that the family of elements

\left(B_{\left|a\right|}(a), e^{\alpha h_{\gamma a}(z)}\right), \quad a \in \mathbb {C} \backslash \left\{ 0\right\}

where the function $h_{\gamma a}$ is defined in Problem 6, forms the function $z^{\alpha }$ , analytic in $\mathbb {C} \backslash \left\{ 0\right\}$ .

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Problem 20.8

For every point $a \in \mathbb {C}, a \neq i, a \neq -i$ , and every piecewise-smooth oriented curve $\gamma_{a}$ starting at the point 0 and ending at the point $a$ , such that $i \notin \gamma_{a},-i \notin \gamma_{a}$ , let us define the quantity

g_{\gamma _{a}}(a) = \int _{\gamma _{a}} \frac{d \zeta }{1+\zeta ^{2}}

and the function $g_{\gamma_{a}}: B_{r_{a}}(a) \rightarrow \mathbb {C}$ of the form

g_{\gamma _{a}}(z) = g_{\gamma _{a}}(a)+\int _{a}^{z} \frac{d \zeta }{1+\zeta ^{2}}, \quad z \in B_{r_{a}}(a), \quad r_{a} = \min \left\{ \left|a-i\right|,\left|a+i\right|\right\}

Prove that the family of elements

\left(B_{r_{a}}(a), g_{\gamma _{a}}\right), \quad a \neq \pm i

forms a function (denoted by $\operatorname {Arctg} z)$ , analytic in $\overline{\mathbb {C}} \backslash (\left\{ i\right\} \cup \left\{ -i\right\} )$ .

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Problem 20.9

Let the function $\varphi (z)$ be regular in a domain $G$ and let $z_{0}$ be some point of this domain. For every point $a \in G$ and every piecewise-smooth curve $\gamma_{a} \subset G$ starting at the point $z_{0}$ and ending at the point $a$ , define the quantity

f_{\gamma _{a}}(a) = \int _{\gamma _{a}} \varphi (\zeta ) d \zeta

and the function $f_{\gamma_{a}}: B_{r}(a) \rightarrow \mathbb {C}$ of the form

f_{\gamma _{a}}(z) = f_{\gamma _{a}}(a)+\int _{a}^{z} \varphi (\zeta ) d \zeta , \quad z \in B_{r_{a}}(a)

where $r_{a} > 0$ is such that $B_{r_{a}}(a) \subset G$ . Prove that the family of elements

\left(B_{r_{a}}(a), f_{\gamma _{a}}\right), \quad a \in G,

forms a function analytic in the domain $G$ .

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Problem 20.10

Prove that the set of values taken by the elements of the analytic function $\operatorname {Ln} z$ (see Problem 6) at an arbitrary point $a \in \mathbb {C} \backslash \left\{ 0\right\}$ coincides with the set of values of the expression

\operatorname {Ln} a = \ln \left|a\right|+i \operatorname {Arg} a.

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Problem 20.11

Prove that the set of values taken by the elements of the analytic function $z^{\alpha }$ (see Problem 7) at an arbitrary point $a \in \mathbb {C} \backslash \left\{ 0\right\}$ coincides with the set of values of the expression

\left\{ a^{\alpha }\right\} = e^{\alpha \operatorname {Ln} a}.

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Problem 20.12

Prove that the set of values taken by the elements of the analytic function $\operatorname {Arctg} z$ (see Example 8) at an arbitrary point $a$ of the domain $G = \left\{ z \in \mathbb {C}: z \neq \pm i\right\}$ coincides with the set of values of the expression

\frac{1}{2 i} \operatorname {Ln} \frac{1+i a}{1-i a}.

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Problem 20.13

For the following multivalued expressions and domains $G$ , find the functions analytic in the domain $G$ that are represented by these expressions:

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(1)

$\left\{ z^{z}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0\right\}$ ;

(2)

$\left\{ z^{2} \operatorname {Ln}^{2} z\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0\right\}$ ;

(3)

$\operatorname {Ln}\left(1-z^{2}\right)$ , where $G = \left\{ z \in \mathbb {C}: z \neq 1, z \neq -1\right\}$ ;

(4)

$\left\{ \sqrt[3]{1+z^{2}}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq i, z \neq -i\right\}$ ;

(5)

$\left\{ \sqrt{1-z^{4}}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq \pm i, z \neq \pm 1\right\}$ ;

(6)

Ln $\frac{z^{2}-1}{z^{2}+1}$ , where $G = \left\{ z \in \overline{\mathbb {C}}: z \neq \pm i, z \neq \pm 1\right\}$ .

(7)

$\left\{ \sqrt{z^{2}-1}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq \pm 1\right\}$ ;

(8)

$\operatorname {Ln} \frac{z-1}{z+1}$ , where $G = \left\{ z \in \mathbb {C}: z \neq \pm 1\right\}$ .

Problem 20.14

(1)(2)

Let the function $f: G \rightarrow \mathbb {C}$ be regular and nonzero in the domain $G$ .

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(1)

Must the expression $\left\{ \sqrt{f(z)}\right\}$ represent a function analytic in the domain $G$ ?

(2)

Can the expression $\left\{ \sqrt{f(z)}\right\}$ represent a function analytic in the domain $G$ ?

Problem 20.15

Prove that if the initial elements of two functions analytic in a domain $G$ are equivalent, then these functions are identically equal.

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Problem 20.16

Prove that if some element of a function analytic in a domain $G$ is equivalent to zero, then this function is identically equal to zero.

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Problem 20.17

Let the initial element $(B_{r}(a), f_{0})$ generate an analytic function in a domain $G \subset \mathbb {C}$ . Prove that in the domain $G$ there exists an analytic function with initial element

\left(B_{r}(a), \int _{a}^{z} f_{0}(\zeta ) d \zeta \right)

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Problem 20.18

Prove that there exists a function analytic in the entire complex plane with the points $z = 0$ and $z = 1$ deleted, which is represented by the formula $\operatorname {LnLn} z$ .

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Problem 20.19

Prove the existence of functions represented by the following multivalued formulas and analytic in the given domains $G$ :

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(1)

$\left\{ (z+1)^{\alpha }(\operatorname {Ln} z)^{\beta }\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq \pm 1\right\}$ ;

(2)

$\operatorname {Ln}\left(z+\sqrt{z^{2}+1}\right)$ , where $G = \left\{ z \in \mathbb {C}: z \neq \pm i\right\}$ ;

(3)

$\operatorname {Ln}(1-\sqrt[4]{z})$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq 1\right\}$ ;

(4)

$\left\{ \sqrt{z+\sqrt{z}}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq 1\right\}$ ;

(5)

$\left\{ \sqrt[3]{\pi i+\operatorname {Ln} z}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq -1\right\}$ ;

(6)

$\left\{ \sqrt{1+\sqrt{\operatorname {Ln} z}}\right\}$ , where $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq e\right\}$ .

Problem 20.20

Prove that the sum of the power series $\sum_{n = 0}^{\infty } z^{2^{n}}$ is a complete analytic function in the disk of convergence $B_{1}(0) = \left\{ z:\left|z\right| < 1\right\}$ .

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Problem 20.21

Prove that when the radius of convergence of the power series $\sum_{n = 0}^{+\infty } a_{n} z^{n}$ equals one and all $a_{n} \geqslant 0$ , then such a series cannot be continued to the point $z = 1$ (Pringsheim's theorem).

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Problem 20.22

Prove that the function $f$ , defined by the series

\sum _{n = 0}^{+\infty } \frac{a z^{n}}{(a-z)^{n+1}}, \quad a \in \mathbb {C} \backslash \left\{ 0\right\}

in its disk of convergence is the direct analytic continuation of the function $g$ , defined by the series $\sum_{n = 0}^{+\infty }\left(\frac{2 z}{a}\right)^{n}$ in its disk of convergence.

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Problem 20.23

Let the function $f$ be regular in a neighborhood of the point $z = 0$ , and let the sequence $\left\{ f^{(n)}(z)\right\}$ converge there uniformly, and

\lim _{n \rightarrow \infty } f^{(n)}(0) = 1

Prove that the limit function of this sequence can be analytically continued into $\mathbb {C}$ .

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