Complex Analysis

If $a$ and $b$ are complex numbers and $a \neq 0$, the set $a^{b}$ consists of those complex numbers $c$ having a logarithm of the form $b\alpha$, for some logarithm $\alpha$ of $a$. (That is, $e^{b\alpha } = c$ and $e^{\alpha } = a$ for some complex number $\alpha$.) Describe set $a^{b}$ when $a = 1$ and $b = 1/3+i$.

Write all values of $i^{i}$ in the form $a+bi$.

Show that a necessary and sufficient condition for three points $a$, $b$, and $c$ in the complex plane to form an equilateral triangle is that

Let the points $a$, $b$, and $c$ lie on the unit circle of the complex plane and satisfy $a+b+c = 0$. Prove that $a$, $b$, and $c$ form the vertices of an equilateral triangle.

Prove that for each integer $n \geq 0$ there is a polynomial $T_{n}(x)$ with integer coefficients such that the identity

holds for all $z$.

Let $z_{1}, z_{2}, \ldots , z_{n}$ be complex numbers. Prove that there exists a subset $J \subset \left\{ 1,2,\ldots ,n\right\}$ such that

Let $a_{1}, a_{2}, \ldots , a_{n}$ be complex numbers. Prove that there is a point $x$ in $[0,1]$ such that

Let $a$ and $b$ be complex numbers whose real parts are negative or $0$. Prove the inequality $\left|e^{a}-e^{b}\right| \leq \left|a-b\right|$.

Let $A$ be a finite subset of the unit disc in the plane, and let $N(A,r)$ be the set of points at distance $\leq r$ from $A$, where $0<r<1$. Show that the length of the boundary $N(A,r)$ is, at most, $C/r$ for some constant $C$ independent of $A$.

For complex numbers $\alpha_{1}, \alpha_{2}, \ldots , \alpha_{k}$, prove

Note: See also Problem 1.3.1.

Show that

for $\left|z\right|<1$.

Suppose the coefficients of the power series

are given by the recurrence relation

Find the radius of convergence of the series and the function to which it converges in its disc of convergence.

Show that the differential equation

has an unique entire solution in the complex plane.

Describe the region in the complex plane where the infinite series

converges. Draw a sketch of the region.

Let $f$ be an analytic function such that

Define a sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ by

What is the radius of convergence of the series

Let the sequence $a_{0}, a_{1}, \ldots$ be defined by the equation

Find

Suppose the power series

converges for $\left|z\right|<R$ where $z$ and the $a_{n}$ are complex numbers. If $b_{n} \in \mathbb {C}$ are such that $\left|b_{n}\right| < n^{2}\left|a_{n}\right|$ for all $n$, prove that

converges for $\left|z\right|<R$.

Let $b_{1}, b_{2}, \ldots$ be a sequence of real numbers such that $b_{k} \geq b_{k+1}$ for all $k$ and $\lim_{k\to \infty } b_{k}=0$. Prove that the power series $\sum_{k=1}^{\infty } b_{k} z^{k}$ converges for all complex numbers $z$ such that $\left|z\right| \leq 1$ and $z \neq 1$.

For which $z \in \mathbb {C}$ does

converge?

Show that

converges for all complex numbers $z$ exterior to the lemniscate

Determine the complex numbers $z$ for which the power series

and its term by term derivatives of all orders converge absolutely.

Suppose

has radius of convergence $R>0$. Show that

is entire and that for $0<r<R$, there is a constant $M$ such that

Let the power series $\sum_{n=0}^{\infty } c_{n} z^{n}$, with positive radius of convergence $R$, represent the function $f$ in the disk $\left|z\right|<R$. For $k=0,1,\ldots$ let $s_{k}$ be the $k$th partial sum of the series $s_{k}(z) = \sum_{n=0}^{k} c_{n} z^{n}$. Prove that

for each $z$ in the disk $\left|z\right|<R$.

Let $R>1$ and let $f$ be analytic on $\left|z\right|<R$ except at $z=1$, where $f$ has a simple pole. If

is the Maclaurin series for $f$, show that $\lim_{n\to \infty } a_{n}$ exists.

Find the radius of convergence $R$ of the Taylor series about $z=1$ of the function $f(z) = 1/(1+z^{2}+z^{4}+z^{6}+z^{8}+z^{10})$. Express the answer in terms of real numbers and square roots only.

Prove that the uniform limit of a sequence of complex analytic functions is complex analytic. Is the analogous theorem true for real analytic functions?

Let $g_{n}(z)$ be an entire function having only real zeros, $n=1,2,\ldots$. Suppose

uniformly on compact sets in $\mathbb {C}$, with $g$ not identically zero. Prove that $g(z)$ has only real zeros.

Let $f, g_{1}, g_{2}, \ldots$ be entire functions. Assume that

Find a Laurent series that converges in the annulus $1<\left|z\right|<2$ to a branch of the function $\log \left(\dfrac {z(2-z)}{1-z}\right)$.

Consider the following four types of transformations:

Here, $z$ is a variable complex number and the other letters denote constant complex numbers. Show that each transformation takes circles to either circles or straight lines.

Give examples of conformal maps as follows:

A fractional linear transformation maps the annulus $r<\left|z\right|<1$ (where $r>0$) onto the domain bounded by the two circles $\left|z-\frac{1}{4}\right| = \frac{1}{4}$ and $\left|z\right| = 1$. Find $r$.

Does there exist an analytic function mapping the annulus

onto the annulus

and taking $C_{1} \to C_{1}$, $C_{4} \to C_{2}$, where $C_{r}$ is the circle of radius $r$?

Exhibit a conformal map from $\left\{ z \in \mathbb {C} \mid \left|z\right|<1, \mathrm{Re}\left(z\right)>0\right\}$ onto $\mathbb {D} = \left\{ z \in \mathbb {C} \mid \left|z\right|<1\right\}$.

Find a one-to-one conformal map of the semidisc

onto the upper half-plane.

Conformally map the region inside the disc given by $\left\{ z \in \mathbb {C} \mid \left|z-1\right| \leq 1\right\}$ and outside the disc $\left\{ z \in \mathbb {C} \mid \left|z-\frac{1}{2}\right| \leq \frac{1}{2}\right\}$ onto the upper half-plane.

Prove that there is no one-to-one conformal map of the punctured disc $G = \left\{ z \in \mathbb {C} \mid 0<\left|z\right|<1\right\}$ onto the annulus $A = \left\{ z \in \mathbb {C} \mid 1<\left|z\right|<2\right\}$.

Let $n$ be a positive integer. Find a group of linear fractional transformations of $\mathbb {C}$ that fixes the two points $z = 1$ and $z = -1$ and has order $n$.

Let $\alpha$ be a number in $(0, \pi /2)$. Prove that the function $f(z) = e^{-1/z}$ is uniformly continuous in $S_{\alpha } = \left\{ z \mid 0<\left|z\right| \leq 1, \left|\operatorname {Arg} z\right| \leq \alpha \right\}$, a sector of the complex plane.

Let $a$ and $b$ be nonzero complex numbers and $f(z) = az+bz^{-1}$. Determine the image under $f$ of the unit circle $\left\{ z \mid \left|z\right| = 1\right\}$.

Let $f$ be analytic on and inside the unit circle $C = \left\{ z \mid \left|z\right| = 1\right\}$. Let $L$ be the length of the image of $C$ under $f$. Show that $L \geq 2\pi \left|f'(0)\right|$.

Let

be analytic in the disc $\mathbb {D} = \left\{ z \in \mathbb {C} \mid \left|z\right|<1\right\}$. Assume $f$ maps $\mathbb {D}$ one-to-one onto a domain $G$ having area $A$. Prove

Compute the area of the image of the unit disc $\left\{ z \mid \left|z\right|<1\right\}$ under the map $f(z) = z+z^{2}/2$.

Let

be an analytic function in the open unit disc $\mathbb {D}$. Assume that

Prove that $f$ is injective.

Let $f(z)$ be a function that is analytic in the unit disc $\mathbb {D} = \left\{ \left|z\right|<1\right\}$. Suppose that $\left|f(z)\right| \leq 1$ in $\mathbb {D}$. Prove that if $f(z)$ has at least two fixed points $z_{1}$ and $z_{2}$, then $f(z) = z$ for all $z \in \mathbb {D}$.

For each $k>0$, let $X_{k}$ be the set of analytic functions $f(z)$ on the open unit disc $\mathbb {D}$ such that

is finite. Show that $f \in X_{k}$ if and only if $f' \in X_{k+1}$.

Let the function $f$ be analytic in the open unit disc of the complex plane and real valued on the radii $[0,1]$ and $[0, e^{i\pi \sqrt{2}}]$. Prove that $f$ is constant.

Let the function $f$ be analytic in the disc $\left|z\right|<1$ of the complex plane. Assume that there is a positive constant $M$ such that

Prove that

Suppose $h(z)$ is analytic in the whole plane, $h(0) = 3+4i$, and $\left|h(z)\right| \leq 5$ if $\left|z\right|<1$. What is $h'(0)$?

Let $f$ be analytic in the closed unit disc, with $f(-\log 2) = 0$ and $\left|f(z)\right| \leq \left|e^{z}\right|$ for all $z$ with $\left|z\right| = 1$. How large can $\left|f(\log 2)\right|$ be?

Suppose that $f$ is analytic on the open upper half-plane and satisfies $\left|f(z)\right| \leq 1$ for all $z$, $f(i) = 0$. How large can $\left|f(2i)\right|$ be under these conditions?

Let $f(z)$ be analytic on the right half-plane $H = \left\{ z \mid \mathrm{Re}\left(z\right)>0\right\}$ and suppose $\left|f(z)\right| \leq 1$ for $z \in H$. Suppose also that $f(1) = 0$. What is the largest possible value of $\left|f'(1)\right|$?

Let $f(z)$ be analytic on the open unit disc $\mathbb {D} = \left\{ z \mid \left|z\right|<1\right\}$. Prove that there is a sequence $(z_{n})$ in $\mathbb {D}$ such that $\left|z_{n}\right| \to 1$ and $(f(z_{n}))$ is bounded.

Let $f$ be an analytic function in the unit disc, $\left|z\right|<1$.

Let $f$ be a holomorphic map of the unit disc $\mathbb {D} = \left\{ z \mid \left|z\right|<1\right\}$ into itself, which is not the identity map $f(z) = z$. Show that $f$ can have, at most, one fixed point.

If $f(z)$ is analytic in the open disc $\left|z\right|<1$, and $\left|f(z)\right| \leq 1/(1-\left|z\right|)$, show that

Let the function $f$ be analytic in the unit disc, with $\left|f(z)\right| \leq 1$ and $f(0) = 0$. Assume that there is a number $r$ in $(0,1)$ such that $f(r) = f(-r) = 0$. Prove that

Let $f(z)$ be an analytic function that maps the open disc $\left|z\right|<1$ into itself. Show that $\left|f'(z)\right| \leq 1/(1-\left|z\right|^{2})$.

Let $f$ be an analytic function in the open unit disc of the complex plane such that $\left|f(z)\right| \leq C/(1-\left|z\right|)$ for all $z$ in the disc, where $C$ is a positive constant. Prove that $\left|f'(z)\right| \leq 4C/(1-\left|z\right|)^{2}$.

Let $f$ be an entire function such that $\mathrm{Re}\left(f(z)\right) \geq -2$ for all $z \in \mathbb {C}$. Show that $f$ is constant.

Let the function $f$ be analytic in the entire complex plane, and suppose that $f(z)/z \to 0$ as $\left|z\right| \to \infty$. Prove that $f$ is constant.

Let the rational function $f$ in the complex plane have no poles for $\mathrm{Im}\left(z\right) \geq 0$. Prove that

Let $f$ be an entire function such that, for all $z$, $\left|f(z)\right| = \left|\sin z\right|$. Prove that there is a constant $C$ of modulus $1$ such that $f(z) = C\sin z$.

Suppose $f$ and $g$ are entire functions with $\left|f(z)\right| \leq \left|g(z)\right|$ for all $z$. Prove that $f(z) = cg(z)$ for some constant $c$.

Let $p$ and $q$ be nonconstant complex polynomials of the same degree whose zeros lie in the open disk $\left|z\right|<1$. Prove that if $\left|p(z)\right| = \left|q(z)\right|$ for $\left|z\right| = 1$ then $q = \lambda p$ for a unimodular constant $\lambda$.

Let $f$ be an entire function. Define $\Omega = \mathbb {C} \setminus (-\infty , 0]$, the complex plane with the ray $(-\infty , 0]$ removed. Suppose that for all $z \in \Omega$, $\left|f(z)\right| \leq \left|\log z\right|$, where $\log z$ is the principal branch of the logarithm. What can one conclude about the function $f$?

Let $f$ and $g$ be two entire functions such that, for all $z \in \mathbb {C}$, $\mathrm{Re}\left(f(z)\right) \leq k\mathrm{Re}\left(g(z)\right)$ for some real constant $k$ (independent of $z$). Show that there are constants $a$, $b$ such that

Let the continuous real-valued function $\varphi$ on the complex plane satisfy the Sub--Mean--Value Property: for any point $z_{0}$,

Prove that $\varphi$ obeys the Maximum Modulus Principle: the maximum of $\varphi$ over any compact set $K$ is attained on the boundary of $K$.

Let $f : \mathbb {C} \to \mathbb {C}$ be an entire function and let $a>0$ and $b>0$ be constants.

Let the function $f$ be analytic in the entire complex plane and satisfy

for all $r>0$. Prove that $f$ is the zero function.

Does there exist a function $f$, analytic in the punctured plane $\mathbb {C} \setminus \left\{ 0\right\}$, such that

for all nonzero $z$?

Let the function $f$ be analytic in the entire complex plane and satisfy the inequality $\left|f(z)\right| \leq \left|\mathrm{Re}\left(z\right)\right|^{-1/2}$ off the imaginary axis. Prove that $f$ is constant.

Let the nonconstant entire function $f$ satisfy the conditions

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.)