Analytic and Meromorphic Functions

Let $f = u+iv$ be analytic in a connected open set $D$, where $u$ and $v$ are real valued. Suppose there are real constants $a$, $b$ and $c$ such that $a^{2}+b^{2} \neq 0$ and

in $D$. Show that $f$ is constant in $D$.

True or false: A function $f(z)$ analytic on $\left|z-a\right|<r$ and continuous on $\left|z-a\right| \leq r$ extends, for some $\delta >0$, to a function analytic on $\left|z-a\right|<r+\delta$? Give a proof or a counterexample.

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

Let $k$ be an integer larger than $1$. Find all entire functions $f$ that satisfy $f(z^{k}) = (f(z))^{k}$.

Let $F$ be a polynomial over $\mathbb {C}$ of positive degree $d$, and let $S$ be its zero set. Prove that every rational function $R$ whose finite poles lie in $S$ can be written uniquely as

where $m$ and $n$ are integers, $m \leq n$, and the $a_{k}$ are polynomials whose degrees are less than $d$.

Let $A = \left\{ 0\right\} \cup \left\{ 1/n \mid n \in \mathbb {Z}, n>1\right\}$, and let $\mathbb {D}$ be the open unit disc in the complex plane. Prove that every bounded holomorphic function on $\mathbb {D} \setminus A$ extends to a holomorphic function on $\mathbb {D}$.

Let $U$ be a connected and simply connected open subset of the complex plane, and let $f$ be a holomorphic function on $U$. Suppose $a$ is a point of $U$ such that the Taylor series of $f$ at $a$ converges on an open disc $D$ that intersects the complement of $U$. Does it follow that $f$ extends to a holomorphic function on $U \cup D$? Give a proof or a counterexample.

Let $f : \mathbb {C} \to \mathbb {C}$ be a nonconstant entire function. Prove that $f(\mathbb {C})$ is dense in $\mathbb {C}$.

Let $L$ be a line in $\mathbb {C}$, and let $f$ be an entire function such that $f(\mathbb {C}) \cap L = \emptyset$. Prove that $f$ is constant. (Do not use the theorem of Picard that the image of a nonconstant entire function omits at most one complex number.)

Let $s(y)$ and $t(y)$ be real differentiable functions of $y$, $-\infty <y<\infty$, such that the complex function

is complex analytic with $s(0) = 1$ and $t(0) = 0$. Determine $s(y)$ and $t(y)$.

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

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

Assume the nonconstant entire function $f$ takes real values on two intersecting lines in the complex plane. Prove that the measure of either angle formed by the lines is a rational multiple of $\pi$.

Let $\Omega$ be an open subset of $\mathbb {R}^{2}$, and let $f : \Omega \to \mathbb {R}^{2}$ be a smooth map. Assume that $f$ preserves orientation and maps any pair of orthogonal curves to a pair of orthogonal curves. Show that $f$ is holomorphic.

Note: Here we identify $\mathbb {R}^{2}$ with $\mathbb {C}$.

Let $f$ be a complex valued function in the open unit disc, $\mathbb {D}$, of the complex plane such that the functions $g = f^{2}$ and $h = f^{3}$ are both analytic. Prove that $f$ is analytic in $\mathbb {D}$.

Prove or supply a counterexample: If $f$ is a continuous complex valued function defined on a connected open subset of the complex plane and if $f^{2}$ is analytic, then $f$ is analytic.

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 the entire function $f$ be real valued on the lines $\mathrm{Im}\left(z\right) = 0$ and $\mathrm{Im}\left(z\right) = \pi$. Prove that $f$ has $2\pi i$ as a period: $f(z+2\pi i) = f(z)$ for all $z$.

Let $f(z)$ be analytic for $z \neq 0$, and suppose that $f(1/z) = f(z)$. Suppose also that $f(z)$ is real for all $z$ on the unit circle $\left|z\right| = 1$. Prove that $f(z)$ is real for all real $z \neq 0$.

Let $p$ be a nonconstant complex polynomial whose zeros are all in the half-plane $\mathrm{Im}\left(z\right)>0$.

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 the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove $f'(x)>0$ for $x$ real.

Let $f = u+iv$ and $g = p+iq$ be analytic functions defined in a neighborhood of the origin in the complex plane. Assume $\left|g'(0)\right|<\left|f'(0)\right|$. Prove that there is a neighborhood of the origin in which the function $h = f+\overline{g}$ is one-to-one.

Prove or disprove: If the function $f$ is analytic in the entire complex plane, and if $f$ maps every unbounded sequence to an unbounded sequence, then $f$ is a polynomial.

Determine the group $\operatorname {Aut}(\mathbb {C})$ of all one-to-one analytic maps of $\mathbb {C}$ onto $\mathbb {C}$.

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 $z$.

Let the function $f$ be analytic in the punctured disc $0<\left|z\right|<r_{0}$, with Laurent series

Assume there is a positive number $M$ such that

Prove that $c_{n} = 0$ for $n<-2$.

Let $a>0$. Show that the complex function

satisfies $\left|f(z)\right|<1$ for all $z$ in the open left half-plane $\mathrm{Re}\left(z\right)<0$.

Let $f$ be a continuous real valued function on $[0,1]$, and let the function $h$ in the complex plane be defined by

Let $f$ be a continuous complex valued function on $[0,1]$, and define the function $g$ by

Prove that $g$ is analytic in the entire complex plane.

Let $f$ and $g$ be analytic functions in the open unit disc, and let $C_{r}$ denote the circle with center $0$ and radius $r$, oriented counterclockwise.

Let $F$ be a continuous complex valued function on the interval $[0,1]$. Let

for $z$ a complex number not in $[0,1]$.

Let $f(z)$ be a meromorphic function on the complex plane. Suppose that for every polynomial $p(z) \in \mathbb {C}^{z}$ and every closed contour $\Gamma$ avoiding the poles of $f$, we have

Prove that $f(z)$ is entire.