Singular Points of Complete Analytic Functions

Find and examine the singular points of the complete analytic functions:

The functions given below are analytic in the annulus $G = \left\{ z: 0 < \left|z\right| < 1\right\}$. Determine the character of the branch point $z = 0$.

Find and examine the singular points of the complete analytic functions:

The functions indicated below are analytic in the plane cut along the segment $[-1,1]$. Determine the character of the singular point $z = \infty$.

Let $z = a$ be a branch point of finite order for the functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$. Prove that for the functions $\mathcal{F}(z)+\mathcal{G}(z)$ and $\mathcal{F}(z) \mathcal{G}(z)$ it is also either a branch point of finite order or an isolated singular point of single-valued character.

Let $z = a$ be a branch point of order $m$ for the function $\mathcal{F}(z)$ and a branch point of order $n$ for the function $\mathcal{G}(z)$. Determine the order of the branch point $z = a$ for the functions $\mathcal{F}(z)+\mathcal{G}(z)$ and $\mathcal{F}(z) \cdot \mathcal{G}(z)$, assuming that $m$ and $n$ are coprime.

Solve the preceding problem, assuming that $m$ and $n$ are different but not necessarily coprime.

Find and examine the singular points of the complete analytic functions:

Let the function $\mathcal{F}(z)$ be analytic in the annulus $G = \left\{ z: r < \left|z\right| < R\right\}$, and let $(B_{r}(a), f_{a})$ be some element of the function $\mathcal{F}(z)$ centered at the point $a \in G$. By the symbol $(B_{r}(a),\left\{ f_{a}\right\}_{m})$ denote the element obtained from the element $\left(B_{r}(a), f_{a}\right)$ by continuation along the circle $\gamma = \left\{ z:\left|z\right| = \left|a\right|\right\}$, traversed $m$ times counterclockwise. Prove that for the function $\mathcal{F}(z)$ to be $n$-valued in the annulus $G$ it is necessary and sufficient that $\left\{ f_{a}(z)\right\}_{m} \neq f_{a}(z), z \in B_{r}(a)$, for $m = 1,2, \ldots , n-1$, while $\left\{ f_{a}(z)\right\}_{n} = f_{a}(z), z \in B_{r}(a)$.

Let $z = a$ be a branch point of order $n$ for the function $\mathcal{F}(z)$. Prove that the expression $\mathcal{F}\left(a+\zeta^{n}\right)$ represents $n$ functions having the point $\zeta = 0$ as an isolated singular point of single-valued character.

Prove that if the function $\mathcal{F}(z)$ has the point $z = a$ as a branch point of order $n$, then it can be represented in the form

where the function $f(w)$ has the point $w = 0$ as an isolated singular point of single-valued character.

Let the function $f(z)$ be regular in the disk $\left|z-a\right| < r$, have at the point $z = a$ a zero of order $m$, and have no other zeros. Prove that the expression $\sqrt[n]{f(z)}$ represents:

1. if the numbers $m$ and $n$ are coprime, then a function analytic in the annulus $0 < \left|z-a\right| < r$ and having at the point $z = a$ an algebraic branch point of order $n$;

2. if $m$ is divisible by $n$, then $n$ distinct functions, regular in the disk $\left|z-a\right| < r$;

3. if the numbers $m$ and $n$ have greatest common divisor $p, 1 < p < n$, then $p$ distinct functions, analytic in the annulus $0 < \left|z-a\right| < r$ and having at the point $z = a$ an algebraic branch point of order $n / p$.

Prove that statements а) and в) of Problem 12 remain valid also in the case where the function $f(z)$ has at the point $z = a$ a pole of order $m$ and has neither zeros nor poles at the other points of the disk $\left|z-a\right| < r$.

Prove that a function $\mathcal{F}(z)$ having at the point $z = a$ an algebraic branch point of order $n$ can be represented in the form $\mathcal{F}(z) = f(\sqrt[n]{z-a})$, where the function $f(w)$ is regular at the point $w = 0$ or has a pole there.

Prove that a function $\mathcal{F}(z)$ having at the point $z = a$ an algebraic branch point of order $n$ can be represented in the form

where the functions $g_{k}(z)$ are regular at the point $z = a$ or have poles there.

Let $z = a$ be an algebraic branch point of the function $\mathcal{F}(z)$. Prove that there exists an annulus $0 < \left|z-a\right| < \delta$ in which no element of the function $\mathcal{F}(z)$ vanishes.

Let $z = a$ be an algebraic branch point of the functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$. Prove that for the functions

the point $z = a$ can only be either an algebraic branch point, or a pole, or a removable singular point.

Let $z = a$ be an algebraic branch point of the function $\mathcal{F}(z)$ and $\lim_{z \rightarrow a} \mathcal{F}(z) = b$, and let $w = b$ be an algebraic branch point of the function $\mathcal{G}(w)$. Prove that $z = a$ is an algebraic branch point of the function $\mathcal{G}[\mathcal{F}(z)]$.

Let $z = a$ be an algebraic branch point of the function $\mathcal{F}(z)$ and $\lim_{z \rightarrow a} \mathcal{F}(z) = b$, with $b \neq \infty$, and let the function $f(w)$ be regular at the point $w = b$ with $f^{\prime }(b) \neq 0$. Prove that the point $z = a$ is an algebraic branch point of the same order for the functions $f[\mathcal{F}(z)]$ and $\mathcal{F}(z)$ (if $f^{\prime }(b) = 0$, then the order of the branch point for the function $f[\mathcal{F}(z)]$ may be smaller than for the function $\mathcal{F}(z)$; see Problem 12).

Let $z = a$ be an algebraic branch point of the second order for the function $\mathcal{F}(z)$ and $\lim_{z \rightarrow a} \mathcal{F}(z) = b$, and let $w = b$ be an algebraic branch point of the third order for the function $\mathcal{G}(w)$. What can the order of the branch point $z = a$ turn out to be for the function $\mathcal{G}[\mathcal{F}(z)]$ (indicate all possibilities)?

Solve the preceding problem for arbitrary branching orders $m$ and $n$ (for the functions $\mathcal{F}(z)$ and $\mathcal{G}(w)$ respectively).

Prove that if the point $z = a$ is a logarithmic branch point for the function $\mathcal{F}(z)$ and an algebraic branch point for the function $\mathcal{G}(z) = e^{\mathcal{F}(z)}$, then the limit of $\mathcal{G}(z)$ as $z \rightarrow a$ can equal only either zero or infinity.

Let the function $\mathcal{F}(z)$ be analytic in the annulus $r < \left|z-a\right| < R$. Prove that the expression $\mathcal{F}\left(a+e^{\zeta }\right)$ represents a collection of a finite or infinite number of functions, regular in the strip $\ln r < \mathrm{Re}\left(\zeta \right) < \ln R$.

Let the function $f(z)$ be regular in a domain $G$ bounded by a simple piecewise-smooth curve, and let a curve $\gamma$ lie in the domain $G$, except for its endpoint $\zeta$, located on the boundary of this domain. Prove that if for some $k$

then the point $\zeta$ is a singular point of the function $f(z)$.

Using the result of Problem 24, prove that all regular branches of the functions given below in the domain $G$ have the point $\zeta$ as a singular point (the point $\zeta$ and the domain $G$ are indicated in parentheses after the formula for the function).

Prove that all branches, regular in the domain $G$, of the functions given below have a singular point at infinity (the domain $G$ is indicated in parentheses):

Prove that the function $e^{-(\operatorname {Ln} z)^{2}}$ admits the separation of regular branches in the half-plane $\mathrm{Re}\left(z\right) > 0$ and that each of the branches has singular points $z = 0$ and $z = \infty$.

The function $(1+i \sqrt{z})^{-2}$ has two branches, regular in the upper half-plane. Prove that the point $z = \infty$ is a singular point for both branches, while the point $z = -1$ is a singular point for only one of them.

Let $f(z)$ be the regular branch of the function $(1+i \sqrt{z})^{-2}$ defined by the condition $\sqrt{z} > 0$ for $z > 0$. Prove that if the function $f(z)$ is considered only as a function in the upper half-plane, then the point $z = -1$ is a singular point for it, whereas if considered only as a function in the lower half-plane, it is not a singular point.

Prove that the function $f(z) = \int_{0}^{\infty } \frac{e^{-t z}}{\sqrt{1+t^{2}}} d t$ is regular in the half-plane $\mathrm{Re}\left(z\right) > 0$ and that the point $z = 0$ is a singular point for it.

Let the function $\varphi (t)$, continuous and positive for $t \geqslant 0$, satisfy the condition

Prove that the function $f(z) = \int_{0}^{\infty } \varphi (t) e^{-t z} d t$ is regular in the half-plane $\mathrm{Re}\left(z\right) > 0$, and that the point $z = 0$ is a singular point for it.

Let the function $\varphi (t)$ be continuous for $t \geqslant 0$ and satisfy the condition

Prove that the function $f(z) = \int_{0}^{\infty } \varphi (t) e^{-t z} d t$ can be analytically continued to the half-plane $\mathrm{Re}\left(z\right) > -\delta$ with the point $z = i \alpha$ deleted, at which this function has a simple pole with principal part $\frac{1}{z-i \alpha }$.

Prove that the analytic continuation of each of the following functions has no singular points in the entire plane except poles. Find all the poles of these functions and the principal parts at these poles.

Let the function $\varphi (t)$ be continuous and positive on the segment $[0,1]$. Prove that the function $f(z) = \int_{0}^{1} \frac{\varphi (t)}{t-z} d t$, regular for $\mathrm{Im}\left(z\right) > 0$ (and even in the entire plane cut along the segment $[0,1]$), has singular points $z = 0$ and $z = 1$.

Let the function $\varphi (t)$ be regular in some domain containing the segment $[0,1]$. Prove that the function $f(z) = \int_{0}^{1} \frac{\varphi (t)}{t-z} d t$, regular for $\mathrm{Im}\left(z\right) > 0$, has singular points $z = 0$ and $z = 1$.

Let the radius of convergence of the series $f(z) = \sum_{n = 0}^{\infty } a_{n} z^{n}$ equal one, and let all coefficients $a_{n}$ be real. Denote $S_{n} = \sum_{k = 0}^{n} a_{k}$. Prove that if $S_{n} \rightarrow +\infty$ or $S_{n} \rightarrow -\infty$, then the point $z = 1$ is singular.

Prove that if on the circle of convergence of the series $f(z) = \sum_{n = 0}^{\infty } a_{n} z^{n}$ there is at least one pole of $f(z)$, then the series diverges at all points of this circle.

Let the sequence $\left\{ c_{n}\right\} , n = 0,1,2, \ldots$ satisfy the condition

Prove that the function $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n}$ can be analytically continued into the disk $\left|z\right| < R$ with the point $z = e^{i \alpha }$ deleted, and at the point $z = e^{i \alpha }$ the function $f(z)$ has a simple pole with principal part $A\left(z-e^{i \alpha }\right)^{-1}$.

Prove that if the function $f(z)$ is regular in the disk $\left|z\right| \leqslant R$, except for a simple pole at the point $R e^{i \alpha }$, with principal part $\frac{A}{z-R e^{i \alpha }}$, then

where

Let the series $f(z) = \sum_{n = 0}^{\infty } a_{n} z^{n}$ have on the boundary of the disk of convergence only one singular point, namely a pole $z = z_{0}$ of order $m$. Prove that $a_{n} = \frac{A n^{m-1}}{z_{0}^{n}}[1+\varepsilon (n)]$, where $\varepsilon (n) \rightarrow 0$ as $n \rightarrow \infty$, and $A$ is a constant, $A \neq 0$.

Prove that if on the boundary of the disk of convergence of the power series $f(z) = \sum_{n = 0}^{\infty } a_{n} z^{n}$ there lies only one singular point $z = z_{0}$, which is a pole for $f(z)$, then the limit exists

Let the radius of convergence of the series $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n}$ equal $R, 0 < R < \infty$. Let us introduce the notation:

Prove that:

1. the function $\nu (r, \varphi )$ is a continuous function of $r$ and $\varphi$ and satisfies the inequality $\nu (r, \varphi ) \leqslant (R-r)^{-1}$;

2. $\max_{0 \leqslant \varphi < 2 \pi } \nu (r, \varphi ) = \frac{1}{R-r}$;

3. if $\nu (r, \varphi ) = \frac{1}{R-r}$, then the point $z = R e^{i \varphi }$ is a singular point of the function $f(z)$;

4. if $\nu (r, \varphi ) < \frac{1}{R-r}$, then the point $z = R e^{i \varphi }$ is not a singular point of the function $f(z)$.

Let $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n}, \varlimsup_{n \rightarrow \infty }\left|c_{n}\right|^{1 / n} = 1$, and let all coefficients $c_{n}$ be nonnegative. Prove that the point $z = 1$ is a singular point of the function $f(z)$.

Prove that the statement of Problem 43 remains valid also under the weaker assumptions:

Let $p$ and $q$ be some positive integers, and let the radius of convergence of the series $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n p+q}$ equal one. Prove that:

1. the function $f(z)$ has at least $p$ singular points on the circle $\left|z\right| = 1$;

2. if $z_{0} = e^{i \alpha }$ is a singular point of the function $f(z)$, then the points

   $$z_{k} = e^{i(\alpha +(2 \pi k / p))}, \quad k = 1,2, \ldots , p-1$$

   are also singular points of it.

Let the radius of convergence of the series $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n}$ equal one and let $c_{n} = 0$ whenever $n$ is divisible by $p$ (here $p$ is a positive integer). Prove that:

1. the function $f(z)$ has at least two singular points on the circle $\left|z\right| = 1$;

2. if $z_{0} = e^{i \alpha }$ is a singular point of the function $f(z)$, then among the points

   $$z_{k} = e^{i(\alpha +(2 \pi k / p))}, \quad k = 1,2, \ldots , p-1$$

   there is at least one more singular point of it.

Prove that every point of the circle $\left|z\right| = 1$ is a singular point for the functions