Isolated Singular Points of Single-Valued Character

Prove that the point $z = a$ is a removable singular point for the following functions:

Prove that the point $z = a$ is a pole of the following functions:

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

Let the functions $f(z)$ and $g(z)$ be regular at the point $z = a$, with $f(a) = g(a) = 0$. Prove that the point $z = a$ is an isolated singular point of single-valued character for the function $F(z) = f(z) / g(z)$, and that it cannot be an essential singular point.

Let $z = a$ be an isolated singular point of single-valued character for the function $f(z)$. Prove that the point $z = a$ is an essential singular point of the function $f(z)$ if and only if there exist two sequences $z_{1}^{\prime }, z_{2}^{\prime }, \ldots$ and $z_{1}^{\prime \prime }, z_{2}^{\prime \prime }, \ldots$ such that $\lim_{n \rightarrow \infty } z_{n}^{\prime } = \lim_{n \rightarrow \infty } z_{n}^{\prime \prime } = a$, while $\lim_{n \rightarrow \infty } f\left(z_{n}^{\prime }\right) = A, \lim_{n \rightarrow \infty } f\left(z_{n}^{\prime \prime }\right) = B, \quad A \neq B$.

Prove that the point $z = \infty$ is an essential singular point for the function $\sin z$.

Prove that the point $z = a$ is an essential singular point of the functions:

Find all isolated singular points of single-valued character for the following functions and determine their type:

Prove that an isolated singular point of single-valued character $z = a$ is a removable singular point for the function $f(z)$, if one of the following conditions holds (arranged in order of decreasing restrictiveness):

Let $z = a$ be an isolated singular point of single-valued character for a function $f(z)$ satisfying, in some neighborhood of this point, the inequality $\left|f(z)\right| < M\left|z-a\right|^{-m}$, where $M$ and $m$ are some positive constants. Prove that the point $z = a$ cannot be an essential singular point of the function $f(z)$.

Let the function $f(z)$ have a pole of order $m$ at the finite point $z = a$. Determine the order of the pole at the point $z = a$ of the function $f^{(n)}(z)$.

Let $P_{m}(z)$ and $Q_{n}(z)$ be polynomials of degree $m$ and $n$, respectively. Find the order of the pole at infinity for the following functions:

Let the functions $f(z)$ and $g(z)$ have, at the point $z = \infty$, poles of orders $m$ and $n$, respectively. Prove that the function $F(z) = f(g(z))$ has, at the point $z = \infty$, a pole of order $m n$.

Let the function $g(z)$ be regular at the point $z = a$ with $g(a) = b$, and let the function $f(\zeta )$ have a pole of order $m$ at the point $\zeta = b$. Prove that the function $F(z) = f(g(z))$ has, at the point $z = a$, a pole of order $m n$, where $n$ is the multiplicity of the zero of the function $g(z)-b$ at the point $z = a$.

Find all singular points of the function $f(z)$ and determine their type, if:

Let a sequence of functions $\left\{ f_{n}(z)\right\}$, regular in the annulus $0 < \left|z-a\right| < r$, converge uniformly on every closed part of this annulus. Prove that if each function $f_{n}(z)$ has a pole of order $m$ at the point $z = a$, then the limit function $f(z)$ is regular in the annulus $0 < \left|z-a\right| < r$ and has, at the point $z = a$, either a pole of order not exceeding $m$, or a removable singular point.

Find all singular points of the function $f(z)$ and determine their type, if:

Let $z = a$ be a pole of the function $f(z)$. Prove that for the function $e^{f(z)}$ the point $z = a$ is an essential singular point.

Let $f(z)$ be a function, regular on the set $\overline{\mathbb {C}}$, except for a single simple pole $z = a$. Prove that if $a = \infty$, then $f(z)$ is a linear function, i.e., $f(z) = A z+B$, where $A \neq 0$, and if $a \neq \infty$, then $f(z)$ is a fractional-linear function, i.e., $f(z) = \frac{a z+b}{c z+d}$, where $a d-b c \neq 0$, $c \neq 0$.

Find all singular points of the function $f(z)$ and determine their type, if:

Let $f(z)$ be an entire function taking each finite value only once. Prove that $f(z)$ is a linear function, i.e., $f(z) = a z+b$, $a \neq 0$.

Let $z = a$ be an essential singular point of the function $f(z)$. Can one assert that for the function $\frac{1}{f(z)}$ the point $z = a$ is also essential?

Let the point $z = a$ be an essential singular point of the function $f(z)$. Denote $M(\rho ) = \max_{\left|z-a\right| = \rho }\left|f(z)\right|$. Prove that for any $k > 0$

Let the function $g(z)$ be regular at the point $z = a$ with $g(a) = b$, and let $\zeta = b$ be an essential singular point of the function $f(\zeta )$. Prove that $z = a$ is an essential singular point of the function $F(z) = f(g(z))$.

Let the functions $a_{1}(z), \ldots , a_{n}(z)$ be regular at the point $a$ or have a pole there, and let $z = a$ be an essential singular point of the function $f(z)$. Prove that $z = a$ is an essential singular point of the function

Find all singular points of the function $f(z)$ and determine their type, if:

Let $P_{m}(z)$ and $Q_{n}(z)$ be polynomials of degree $m$ and $n$, respectively, having no common zeros. Prove that the zeros of the polynomial $Q_{n}(z)$, and only these points, are poles of the function $f(z) = \frac{P_{m}(z)}{Q_{n}(z)}$, and the function $f(z)$ has no other singular points in the finite plane. Show that the point $z = \infty$ is a pole of order $m-n$ of the function $f(z)$ if $m > n$, and a point of regularity if $m \leqslant n$.

Let a meromorphic function $f(z)$ have, in $\overline{\mathbb {C}}$, only a finite number of poles $a_{1}, a_{2}, \ldots , a_{m}$ (the point $z = \infty$ may also be a pole). Prove that $f(z)$ is a rational function (a ratio of two polynomials) and is represented in the form

where $f_{k}(z)$ and $f_{0}(z)$ are, respectively, the principal parts of the Laurent series of the function $f(z)$ in the neighborhoods of the points $a_{k}$ and $z = \infty , A = \lim_{z \rightarrow \infty }\left(f(z)-f_{0}(z)\right)$.

Let the function $f(z)$ be regular in the domain $D = \left\{ z: 0 < \left|z\right| < \rho \right\}$ and $f(z) \not\equiv 0$. What conditions must the function $f(z)$ satisfy at the point $z = 0$ so that one can specify a function $g(z)$, regular in the disk $B_{\rho }(0) = \left\{ z:\left|z\right| < \rho \right\}$, with $g(z) \not\equiv 0$, such that $\lim_{z \rightarrow 0}[f(z) g(z)] = 0$.

Let the function $f(z)$ be regular in a punctured neighborhood of the point $z = 0$, i.e., in the domain $D = \left\{ z: 0 < \left|z\right| < \rho \right\}$, and let the inequality

hold for all $z \in D$, where $A > 0$. Prove that $z = 0$ is a removable singular point of the function $f(z)$.

Let an entire function $f(z)$ satisfy, in some domain $\left|z\right| > R$, the condition $\left|f(z)\right| \geqslant A\left|z\right|^{m}$, where $A > 0$ and $m \geqslant 0$ is an integer. Prove that $f(z)$ is a polynomial.