Identity theorem

Do there exist functions $f(z)$ and $g(z)$ that are analytic at $z = 0$ and that satisfy

Determine all the complex analytic functions $f$ defined on the unit disc $\mathbb {D}$ which satisfy

for $n = 2, 3, 4, \ldots$.

Let the function $f$ be analytic in the region $\left|z\right|>1$ of the complex plane. Prove that if $f$ is real valued on the interval $(1, \infty )$ of the real axis, then $f$ is also real valued on the interval $(-\infty , -1)$.

Let the function $f$ be analytic in the complex plane, real on the real axis, $0$ at the origin, and not identically $0$. Prove that if $f$ maps the imaginary axis into a straight line, then that straight line must be either the real axis or the imaginary axis.

Let $f$ be an analytic function in the connected open subset $G$ of the complex plane. Assume that for each point $z$ in $G$, there is a positive integer $n$ such that the $n$-th derivative of $f$ vanishes at $z$. Prove that $f$ is a polynomial.

Let $f(z)$ be a nonconstant meromorphic function. A complex number $w$ is called a period of $f$ if $f(z+w) = f(z)$ for all $z$.

True-False. If the assertion is true, quote a relevant theorem or reason; if false, give a counterexample or other justification.

Suppose that $u$ and $v$ are real valued harmonic functions on a domain $\Omega$ such that $u$ and $v$ satisfy the Cauchy-Riemann equations on a subset $S$ of $\Omega$ which has a limit point in $\Omega$. Prove that $u + iv$ must be analytic on $\Omega$.

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

Let $f(z)$ be regular and bounded on the disk $\left|z\right|<1$ and let

hold uniformly in a sector $\alpha \leq \vartheta \leq \beta$, $\alpha <\beta$. Then $f(z)$ vanishes identically.

There does not exist a function which is regular analytic along the real axis and which assumes for real values of the variable every value in the interior of a fixed circle. In brief: There does not exist an analytic Peano-curve.

Suppose $f \in H(U), g \in H(U)$, and neither $f$ nor $g$ has a zero in $U$. If

find another simple relation between $f$ and $g$.

If $u$ is a harmonic function in a region $\Omega$, what can you say about the set of points at which the gradient of $u$ is 0? (This is the set on which $u_{x} = u_{y} = 0$).

Suppose $C\left\{ M_{n}\right\}$ is quasi-analytic, $f \in C\left\{ M_{n}\right\}$, and $f(x) = 0$ for infinitely many $x \in [0,1]$. What follows?

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 $G$.

Does there exist a function $f(z)$, regular in some neighborhood of the point $z = 0$ and satisfying one of the following conditions (for all $n = 1,2, \ldots )$:

Let the functions $f_{1}(z)$ and $f_{2}(z)$ be regular in a domain $D$ and satisfy in this domain the differential equation $f^{\prime }(z) = P(z, f(z))$, where $P(z, w)$ is a polynomial in its variables. Prove that if at some point $z_{0} \in D$ the equality $f_{1}\left(z_{0}\right) = f_{2}\left(z_{0}\right)$ holds, then $f_{1}(z) \equiv f_{2}(z)$.

Let the functions $f_{1}(z)$ and $f_{2}(z)$ be regular in a domain $D$ and satisfy in this domain the differential equation

where $P$ is a polynomial in its variables. Prove that if at some point $z_{0} \in D$ the equalities

hold, then $f_{1}(z) \equiv f_{2}(z)$.

Prove that the functional equation $f(z) = f(2 z)$ has no solutions regular at the point $z = 0$ and not identically constant.

Let $q = e^{2 \pi i \alpha }$, where $\alpha$ is an irrational number. Prove that the functional equation $f(z) = f(q z)$ has no solutions $f(z)$ regular for $1 / 2 < \left|z\right| < 2$ and not identically constant.

Let $f(z)$ be a periodic function, regular in some domain containing the point $z = \infty$. Prove that $f(z) =$ const throughout the domain of regularity.

Let the function $\Phi \left(z_{1}, \ldots , z_{n}\right)$ be defined when

and be regular in each variable $z_{k}$ in the domain $D_{k}$ for arbitrary fixed values of the remaining variables. Prove that if each domain $D_{k}$ contains a nonempty interval $\left(a_{k}, b_{k}\right)$ of the real axis, and if

then $\Phi \left(z_{1}, \ldots , z_{n}\right) \equiv 0$.

Relying on the validity of the formulas given below for real values of the variables, prove their validity also for arbitrary complex values of these variables:

Prove that if the function $f(z)$ is regular in a domain $D$ and for each point $z$ of $D$ there exists an $n$ such that $f^{(n)}(z) = 0$, then $f(z)$ is a polynomial.

Prove that if the initial elements of two functions analytic in a domain $G$ are equivalent, then these functions are identically equal.

Prove that if some element of a function analytic in a domain $G$ is equivalent to zero, then this function is identically equal to zero.