Math Problems Vault

Problem 5.6.7 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

Let $U$ be a connected and simply connected open subset of the complex plane, and let $f$ be a holomorphic function on $U$ . Suppose $a$ is a point of $U$ such that the Taylor series of $f$ at $a$ converges on an open disc $D$ that intersects the complement of $U$ . Does it follow that $f$ extends to a holomorphic function on $U \cup D$ ? Give a proof or a counterexample.

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Problem 5.1.13 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.1 "Analytic and Harmonic Functions"

Let $f$ be the analytic function defined in the disk $\Delta = \left\{ z : \left|z-4\right| < 4\right\}$ so that $f(z) = z^{1/3}(z+1)^{1/2}$ in $\Delta$ and $f(x)$ is positive for $0 < x < 8$ . An analytic function $g$ in $\Delta$ is obtained from $f$ by analytic continuation along the path starting and ending at $z = 4$ (see Fig. 5.2). Express $g$ in terms of $f$ .

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Problem 5.1.16 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.1 "Analytic and Harmonic Functions"

(a)(b)

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(a)

Let $D \subset \mathbb {C}$ be the complement of the simply connected closed set $\left\{ e^{\theta +i\theta } : \theta \in \mathbb {R}\right\} \cup \left\{ 0\right\}$ . Let $\log$ be a branch of the logarithm on $D$ such that $\log e = 1$ . Find $\log e^{15}$ . Justify your answer.

(b)

Let $\gamma$ denote the unit circle, oriented counterclockwise. By lifting the integration to an appropriate covering space, give a precise meaning to the integral $\int_\gamma (\log z)^2\, dz$ and find all possible values which can be assigned to it.

Problem 9.10 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 9 "The Uniqueness Theorem. Regular Continuation."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 9 "The Uniqueness Theorem. Regular Continuation."

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 20.6 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 20.18 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 20 "Analytic Continuation. Complete Analytic Functions."

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 $G = \left\{ z \in \mathbb {C}: z \neq 0, z \neq e\right\}$ .

Complete analytic functions defined by multivalued formulas

References