Sequences of Analytic Functions

We use the same notation and hypothesis as before and expand

Derive this formula from Lagrange's formula or Lagrange's formula from this one by correctly using the generality of both formulas. [The existence of one formula for a certain $f(z)$ implies immediately the other formula for another $f(z)$.]

Prove the formula in problem 207 directly by expressing the coefficient of $w^{n}$ as a Cauchy integral.

Expand in ascending powers of $w$ the solution of the transcendental equation

that vanishes for $w=0$.

(continued). Expand $e^{\alpha z}$ in powers of $w=ze^{-z}$ where $\alpha$ is an arbitrary constant.

Expand in ascending powers of $w$ the solution $x$ of the trinomial equation

that becomes $1$ for $w=0$.

(continued). Expand $x^{\alpha }$ in powers of $w$, where $\alpha$ denotes an arbitrary constant. ($x^{\alpha }=y$ is the solution of the trinomial equation

(continued). Note the cases $\beta =0,1,2,-1,\frac{1}{2}$ and derive problems 209, 210 by taking the limit in problems 211, 212.

Evaluate the sum of the power series

What is its radius of convergence?

Prove problem 156 with the help of the results of problem 214.

Let $\alpha$ and $\beta$ denote rational numbers. Then the series

represents an algebraic function of $w$.

Arrange the successive powers of the trinomial $1+w+w^{2}$ in a regular triangular array

The sum of the middle terms (in boldface: $1, w, 3w^{2}, 7w^{3}, \ldots$) is

Arrange the successive powers of the binomial $1+w$ in a regular triangular array (Pascal's triangle)

Find the sum of the middle terms (in boldface) and, more generally, the sum of any column.

Find the generating functions of the polynomials $P_{n}(x)$, $P_{n}^{(\alpha ,\beta )}(x)$, $L_{n}^{(\alpha )}(x)$, defined by the formulas

Let $s$ be a constant of sufficiently small modulus. Then the following formulas are valid for $F(z)=e^{sz}$:

The four formulas mentioned in problem 220 hold for any polynomial $F(z)$. (In this case the series are obviously finite.)

The formulas (1), (2) given in problem 220 are also valid for any rational function $F(z)$ if the real part of $z$ is larger than the real part of any of the finite poles of $F(z)$; formula (1) requires the additional condition that $F(z)$ be regular for $z=0,1,2,3,\ldots$. --Do the formulas (3) and (4) hold for rational but not entire functions?

where $a_{k}$, $k=0,\pm 1,\pm 2,\ldots$, denote arbitrary constants.

Define

and establish the relation

Define

and show that

Define

and show that

$\sin \pi z$ is a single-valued function of $w=z(1-z)$. The expansion of $\sin \pi z$ in powers of $w$ contains only positive coefficients (except the constant term) [problem 227].

Prove that

We assume that the function

Express the coefficients $a_{1},a_{2},\ldots ,a_{n},\ldots$ in terms of the real and the imaginary part resp.of $f(z)$ on the circle $\left|z\right|=r$, $0<r<R$.

(continued). We set $\re f(re^{i\vartheta }) = U(r,\vartheta )$ and assume that $f(0)$ is real and $\left|z\right|<r$. Then

(continued). Suppose $f(z)$ does not vanish on $\left|z\right|=r$ and that its zeros in the disk $\left|z\right|<r$ are $c_{1},c_{2},\ldots ,c_{m}$. Then we have for $\left|z\right|<r$

where $\gamma$ is a real constant. [Consequence of problem 231 as problem 120 is a consequence of problem 119.]

The function $f(z)$ is regular, its real part is positive in the open disk $\left|z\right|<R$ and continuous in the closed disk $\left|z\right|\leq R$. If the real part becomes identically zero on an arc of the circle the imaginary part of $f(z)$ changes on this arc always in the same sense: it decreases as $\arg z$ increases.

Let $f(z) = a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n}+\cdots$ be regular in the disk $\left|z\right|<R$, $f(re^{i\vartheta }) = U(r,\vartheta )+iV(r,\vartheta )$, $U(r,\vartheta ),V(r,\vartheta )$ are real. Then the equation

holds for $0<r<R$ provided that it holds for $r=0$.

The function

is assumed to be regular and to have positive real part in the open disk $\left|z\right|<1$. Then $\left|a_{n}\right|\leq 1$, $n=1,2,\ldots$ In none of these inequalities can we replace $1$ by a smaller number.

Suppose that the function

on the disk $\left|z\right|<R$. Then the inequality

holds for $0<r<R$. Example: $f(z)=\dfrac {z+1}{z-1}$, $R=1$, $A=0$.

We assume that the Laurent series

converges on the annulus $0<\left|z\right|<\infty$ (sphere from which two points have been removed) and that $z=0$ and $z=\infty$ are essential isolated singular points. The maximum of the real part of $\psi (z)$ on the circle $\left|z\right|=r$ is denoted by $A(r)$. $A(r)$ increases to $\infty$ faster than any power of $r$ as $r\to \infty$ and faster than any power of $\dfrac {1}{r}$ as $r\to 0$. Precisely

We assume that the function $f(z) = a_{0}+a_{1}z+\cdots +a_{n}z^{n}+\cdots$ is regular in the disk $\left|z\right|<R$ and we denote by $\Delta (f)$ the largest oscillation of the real part of $f(z)$ in $\left|z\right|<R$, i.e.$\Delta (f)$ is the least upper bound of $\left|\re f(z_{1})-\re f(z_{2})\right|$ for $\left|z_{1}\right|<R$, $\left|z_{2}\right|<R$. Then we have

The factor $\dfrac {2}{\pi }$ cannot be replaced by a smaller one.

Interpret this proposition geometrically.

Let the function $f(z) = a_{0}+a_{1}z+\cdots +a_{n}z^{n}+\cdots$ be regular in the disk $\left|z\right|<R$ and denote by $D(f)$ the largest oscillation of $f(z)$ for $\left|z\right|<R$, i.e.$D(f)$ is the least upper bound of $\left|f(z_{1})-f(z_{2})\right|$ for $\left|z_{1}\right|<R$, $\left|z_{2}\right|<R$. Then we have

The factor $\frac{1}{2}$ can not be replaced by a smaller one. Interpret this proposition geometrically.

Let the function $f(z)$ satisfy the following conditions:

1. $f(z)$ is regular, $\left|f(z)\right|\leq M$ in the disk $\left|z-s\right|\leq r$;

2. $f(z)$ does not vanish in the closed half-disk $\left|z-s\right|\leq r$, $\re (z-s)\geq 0$;

3. $f(z)$ has in the disk $\left|z-s\right|\leq \frac{2}{3}r$ the zeros $c_{1},c_{2},\ldots ,c_{l}$. Then

[We may assume $s=0$; problems 232, 120.]

Suppose that the unit circle is the circle of convergence of a certain power series and that there are only poles of first order on this circle (no other singularities). Then the sequence of coefficients is bounded.

If there is only one singular point $z_{0}$ on the circle of convergence of $\displaystyle \sum_{n=0}^{\infty } a_{n}z^{n}$ and if $z_{0}$ is a pole then

Let $\displaystyle \sum_{n=0}^{\infty } a_{n}z^{n}$ be the expansion into a power series of a rational function whose denominator (relative prime to the numerator) has degree $q$. The radius of convergence is called $\varrho$ and $A_{n}$ denotes the largest among the $q$ numbers $\left|a_{n}\right|,\left|a_{n-1}\right|,\ldots ,\left|a_{n-q+1}\right|$. Then

(lim not lim sup!)

Let $\nu_{n}$ be the number of non-zero coefficients among the $n$ coefficients $a_{0},a_{1},\ldots ,a_{n-1}$. If there are only poles (and no other singularities) on the circle of convergence of the power series $a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n}+\cdots$ the number of such poles is not smaller than

Example: $1+z^{k}+z^{2k}+z^{3k}+\cdots = \dfrac {1}{1-z^{k}}$.

We assume that the coefficients $a_{0},a_{1},\ldots ,a_{n},\ldots$ of the power series $a_{0}+a_{1}z+\cdots +a_{n}z^{n}+\cdots$ are real and that $\varrho e^{i\alpha }$ and $\varrho e^{-i\alpha }$ are poles and the only singularities on the circle of convergence, $0<\alpha <\pi$. We call $V_{n}$ the number of changes of sign in the sequence $a_{0},a_{1},\ldots ,a_{n-1},a_{n}$. Then

If there is also a pole among the singularities on the circle of convergence the power series converges at no point of the circle of convergence.

Suppose that the point $z=1$ is a regular point of the power series

that converges inside the unit disk. Then the Dirichlet series, assumed convergent for certain values of $s$,

defines an entire function. In the case where the point $z=1$ is a pole of order $h$ of the function $f(z)$, $D(s)$ is a meromorphic function which can have poles only at the points $s=1,2,\ldots ,h$ (only the last one must be a pole); the poles are simple. [$D(s)\Gamma (s) = \displaystyle \int_{0}^{\infty } x^{s-1}f(e^{-x})\, dx$, cf.Part II, problem 117; $\displaystyle \int_{\varrho }^{\infty } x^{s-1}f(e^{-x})\, dx$ is an entire function of $s$ if $\varrho >0$.]

The sequence $a_{0},a_{1},a_{2},\ldots ,a_{n},\ldots$ is assumed to satisfy the condition

Then the series

converges in the infinite strip

of the $s$-plane. The convergence is absolute and uniform in any closed interior strip $-h+\varepsilon \leq \mathrm{Re}\left(s\right)\leq h-\varepsilon$, $\varepsilon >0$; there the series defines an analytic function $\Phi (s)$. The function $\Phi (s)$ vanishes identically only if all the coefficients $a_{0},a_{1},a_{2},\ldots ,a_{n},\ldots$ vanish. [Compute

We assume that the power series

converges inside the unit disk and that $f(z)$ and all its derivatives tend to $0$ as $z$ tends to $1$ on the real axis, i.e.$\lim \limits_{z=x\to 1} f^{(n)}(z) = 0$, $n=0,1,2,\ldots$ Then there are two possibilities:

1. $f(z)$ vanishes identically;

2. $z=1$ is a singular point for $f(z)$. If the power series converges in a disk larger than the unit disk, i.e.if

we are necessarily dealing with the first case. The weaker condition

admits the same conclusion. [Construct the function $\Phi (s)$ of problem 248.]

The proposition 249 is not valid if the condition on the coefficients $a_{0},a_{1},a_{2},\ldots ,a_{n},\ldots$ of $f(z)$, namely $\limsup \limits_{n\to \infty }\dfrac {\log \left|a_{n}\right|}{\sqrt{n}}<0$, is replaced by

[Put

If the series

converges at one single point at which $g(z)$ is regular then $g(z)$ is an entire function and the series converges at every point. The convergence is uniform in any finite domain of the $z$-plane.

If the sequence

is bounded at a single point of the $z$-plane then $g(z)$ is an entire function and the sequence stays bounded at all the points of the $z$-plane. It even has the same limit superior at all the points.

Let

be two infinite sequences, the second one being arbitrary, the first one such that $a_{n}\neq 0$, $a_{m}\neq a_{n}$ when $m\neq n$, $m,n=0,1,2,\ldots$ and that

converges absolutely. The equations

define a unique polynomial $Q_{n}(z)$ of degree not larger than $n$. If the sequence

converges at a single point $z$ different from $a_{0},a_{1},a_{2},\ldots$ it converges at every point $z$, even uniformly in any finite domain of the $z$-plane.

Let

denote a sequence that is infinite in two directions and $Q_{2n}(z)$ be the polynomial of degree $2n$ that satisfies the equations

If the sequence of polynomials

converges for two different non-integral values of $z$ it converges at every point $z$, even uniformly in any finite domain of the $z$-plane.

The sequence $c_{0},c_{1},c_{2},\ldots ,c_{n},\ldots$ is given. A polynomial $Q_{n}(z)$ of degree not exceeding $n$ can be found [Part VI, problem 76] for which

These $n+1$ conditions define $Q_{n}(z)$ uniquely [Part VI, problem 75]. If the sequence

converges at a single point $z$, $z\neq 0$, it converges at every point $z$, in fact uniformly in every finite domain of the $z$-plane.

We assume that in the unit disk, $\left|z\right|<1$, the functions $f_{0}(z),f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$ are regular and different from zero and that their absolute values are smaller than $1$. If $\lim \limits_{n\to \infty } f_{n}(0)=0$, then $\lim \limits_{n\to \infty } f_{n}(z)=0$ in the entire open disk $\left|z\right|<1$; the convergence is actually uniform in every smaller disk $\left|z\right|\leq r<1$.

The harmonic functions

are assumed to be regular and positive in a certain open region $\mathfrak {R}$ of the $x,y$-plane. If the infinite series

converges at a single point of $\mathfrak {R}$ it converges everywhere in $\mathfrak {R}$; in fact, it converges uniformly in any closed subdomain of $\mathfrak {R}$.

We suppose that the functions of the sequence $f_{0}(z),f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$ are analytic in the open region $\mathfrak {R}$ and that the sequence of their real parts converges uniformly in every closed subdomain of $\mathfrak {R}$. Then the sequence of their imaginary parts either diverges at all points or it converges uniformly in any closed subdomain of $\mathfrak {R}$.

The series

converges uniformly in any domain that lies either entirely inside or entirely outside the unit circle and its sum is $z$ or $1$ according as $\left|z\right|<1$ or $\left|z\right|>1$ [problem 14 of Part I].

Let $\alpha$ denote an arbitrary constant, $\alpha \neq 0$. The series

converges uniformly for all positive values of $x$; it represents the function $e^{\alpha x}$ for $0\leq x\leq 1$ and a different analytic function for $1<x<\infty$.

The sequence of functions $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$

converges uniformly in any finite domain that does not contain the imaginary axis.

Let

and put

The sequence of iterated functions

converges to $+1$, when $\mathrm{Re}\left(z\right)>0$, to $-1$ when $\mathrm{Re}\left(z\right)<0$ and diverges when $\mathrm{Re}\left(z\right)=0$.

Let $h(\varphi )$ denote the support function of the infinite convex domain considered in problem 114. The sequence $f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots$