Zeros of Holomorphic Functions

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Problem 2.10.15 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.10 "Elementary Properties of Holomorphic Functions"

Suppose $\Omega$ is a region, $\varphi \in H(\Omega ), \varphi^{\prime }$ has no zero in $\Omega , f \in H(\varphi (\Omega )), g = f \circ \varphi , z_{0} \in \Omega$ , and $w_{0} = \varphi \left(z_{0}\right)$ . Prove that if $f$ has a zero of order $m$ at $w_{0}$ , then $g$ also has a zero of order $m$ at $z_{0}$ . How is this modified if $\varphi^{\prime }$ has a zero of order $k$ at $z_{0}$ ?

Problem 7.11 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 7 "Taylor Series"

(1)(2)(3)(4)

Determine the order $m$ of the zero $z = a$ of the function $f(z)$ , if:

(1)

$f(z) = (\cos 3 z-\cos 5 z)^{2}(1-\cos 2 z)^{3}, \quad a = 0$ ;

(2)

$f(z) = \left(z^{2}-\pi^{2}\right)^{3} \sin^{3} z, \quad a = \pi$ ;

(3)

$f(z) = \left(z^{2}+\pi^{2}\right)^{2}\left(e^{2 z}-1\right)^{4}, \quad a = \pi i$ ;

(4)

$f(z) = \left(z^{4}+2 z^{3}-2 z-1\right)^{2}\left(e^{i \pi z}+1\right)^{3}, a = -1$ .

Problem 9.1 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 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 $G$ .

Problem 12.3 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 12 "Isolated Singular Points of Single-Valued Character"

(1)(2)

Let the functions $f(z)$ and $g(z)$ be regular at the point $z = a$ , with $f(a) = g(a) = 0$ . Prove that:

(1)

$\lim_{z \rightarrow a} \frac{f(z)}{g(z)} = \lim_{z \rightarrow a} \frac{f^{\prime }(z)}{g^{\prime }(z)}$ ;

(2)

$\lim_{z \rightarrow a} \frac{f(z)}{g(z)} = \lim_{z \rightarrow a}\left\{ \frac{g^{\prime }(z) f^{2}(z)}{f^{\prime }(z) g^{2}(z)}\right\}$ .

References

[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.