Holomorphic logarithms and roots of nonvanishing functions

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Problem 5.6.17 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

Prove or supply a counterexample: If $f$ is a continuous complex valued function defined on a connected open subset of the complex plane and if $f^{2}$ is analytic, then $f$ is analytic.

Problem 5.6.18 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

(1)(2)

(1)

Let $G$ be an open connected subset of the complex plane, $f$ an analytic function in $G$ , not identically $0$ , and $n$ a positive integer. Assume that $f$ has an analytic $n$ -th root in $G$ ; that is, there is an analytic function $g$ in $G$ such that $g^{n} = f$ . Prove that $f$ has exactly $n$ analytic $n$ -th roots in $G$ .

(2)

Give an example of a continuous real valued function on $[0,1]$ that has more than two continuous square roots on $[0,1]$ .

Problem 5.6.19 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.6 "Analytic and Meromorphic Functions"

(1)(2)

(1)

Prove that if $f$ is holomorphic on the unit disc $\mathbb {D}$ and $f(z) \neq 0$ for all $z \in \mathbb {D}$ , then there is a holomorphic function $g$ on $\mathbb {D}$ such that $f(z) = e^{g(z)}$ for all $z \in \mathbb {D}$ .

(2)

Does the conclusion of Part 1 remain true if $\mathbb {D}$ is replaced by an arbitrary connected open set in $\mathbb {C}$ ?

Problem 5.7.14 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 5.7 "Cauchy's Theorem"

Let $f(z)$ be analytic in the annulus $\Omega = \left\{ 1 < \left|z\right| < 2\right\}$ . Assume that $f$ has no zeros in $\Omega$ . Show that there exists an integer $n$ and an analytic function $g$ in $\Omega$ such that, for all $z \in \Omega$ , $f(z) = z^{n} e^{g(z)}$ .

Problem 2.13.9 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.13 "Approximation by Rational Functions"

Suppose $\Omega$ is a simply connected region, $f \in H(\Omega ), f$ has no zero in $\Omega$ , and $n$ is a positive integer. Prove that there exists a $g \in H(\Omega )$ such that $g^{n} = f$ .

Problem 2.13.10 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.13 "Approximation by Rational Functions"

Suppose $\Omega$ is a region, $f \in H(\Omega )$ , and $f \not\equiv 0$ . Prove that $f$ has a holomorphic logarithm in $\Omega$ if and only if $f$ has holomorphic $n$ th roots in $\Omega$ for every positive integer $n$ .

Problem 20.14 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."

(1)(2)

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

(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 $G$ ?

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.