Complex Analysis

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

Let $G$ be a region in $\mathbb {C}$ and suppose $u : G \to \mathbb {R}$ is a harmonic function.

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 $\Omega$.

Let $Q = [0,1] \times [0,1] \subset \mathbb {C}$ be the unit square, and let $f$ be holomorphic in a neighborhood of $Q$. Suppose that

Show that $f$ is constant.

Let $f$ be continuous on the closure $\overline{S}$ of the unit square

and let $f$ be analytic on $S$. If $\mathrm{Re}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ y = 0\right\} \cup \left\{ y = 1\right\} )$, and if $\mathrm{Im}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ x = 0\right\} \cup \left\{ x = 1\right\} )$, prove that $f \equiv 0$ everywhere on $\overline{S}$.

Let $P(z)$ be a polynomial of degree $d$ with simple roots $z_1, z_2, \ldots , z_d$. A ``partial fractions'' expression of $\dfrac {1}{P}$ has the form:

Suppose $f$ is meromorphic in a neighborhood of $\overline{D}$ ($D = \left\{ \left|z\right| < 1\right\}$) whose only pole is a simple one at $z = a \in D$. If $f(\partial D) \subseteq \mathbb {R}$, show that there is a complex constant $A$ and a real constant $B$ such that

Let $K_1, K_2, \ldots , K_n$ be pairwise disjoint disks in $\mathbb {C}$, and let $f$ be an analytic function in $\mathbb {C} \setminus \bigcup_{j=1}^{n} K_j$. Show that there exist functions $f_1, f_2, \ldots , f_n$ such that

Recall that a divisor $D_f$ of a rational function $f(z)$ on $\mathbb {C}$ is a set $\left\{ p \in \mathbb {C} \cup \left\{ \infty \right\} \right\}$, consisting of zeros and poles $p$ of $f(z)$ (including the point $\infty$), counted with their multiplicities $n_p \in \mathbb {Z}$. Let $f$ and $g$ be two rational functions with disjoint divisors. Prove that

Let $f(z)$ be the ``branch'' of $\log z$ defined off the negative real axis so that $f(1) = 0$.

Let $f$ be the analytic function defined in the disk $\Delta = \left\{ z : \left|z-4\right| < 4\right\}$ so that $f(z) = z^{1/3}(z+1)^{1/2}$ in $\Delta$ and $f(x)$ is positive for $0 < x < 8$. An analytic function $g$ in $\Delta$ is obtained from $f$ by analytic continuation along the path starting and ending at $z = 4$ (see Fig. 5.2). Express $g$ in terms of $f$.

Define

Let $\Omega$ be the plane with the segment $\left\{ -1 \leq x \leq 1, y = 0\right\}$ deleted. For which of the multi-valued functions

Find the most general harmonic function of the form $f(\left|z\right|)$, $z \in \mathbb {C} \setminus 0$. Which of these $f(\left|z\right|)$ have a single valued harmonic conjugate?

Consider the regular pentagram centered at the origin in the complex plane. Let $u$ be the harmonic function in the interior of the pentagram which has boundary values $1$ on the two segments shown and $0$ on the rest of the boundary. What is the value of $u$ at the origin? Justify your claim.

Suppose $G$ is a region in $\mathbb {C}$, $[0,1] \subset G$, and $h : G \to \mathbb {R}$ is continuous. $h|_{G \setminus [0,1]}$ is harmonic, does this imply that $h$ is harmonic on $G$?

Let $\gamma$ be an arc of the unit circle. Suppose that $u$ and $v$ are harmonic in $D = \left\{ z : \left|z\right| < 1\right\}$ and continuously differentiable on $D \cup \gamma$. If the boundary values satisfy $u = v$ on $\gamma$ and the radial derivatives satisfy $\dfrac {\partial u}{\partial r} = \dfrac {\partial v}{\partial r}$ on $\gamma$, prove that $u = v$ in $D$.

Use conformal mapping to find a harmonic function $U(z)$ defined on the unit disc $\left\{ \left|z\right| < 1\right\}$ such that

Give the correct determination of any multiple-valued functions appearing in your answer.

Determine all continuous functions on $\left\{ z \in \mathbb {C} : 0 \leq \left|z\right| \leq 1\right\}$ which are harmonic on $\left\{ z : 0 < \left|z\right| < 1\right\}$ and which are identically $0$ on $\left\{ z \in \mathbb {C} : \left|z\right| = 1\right\}$.

Let $B(z) = \displaystyle \prod_{k=1}^{n} \dfrac {a_k - z}{1 - \overline{a_k} z}$, where $a_1, a_2, \ldots , a_n$ are distinct points in $D \setminus \left\{ 0\right\}$. Show that

Does there exist an analytic function in $\left|z\right| < 1$ such that $0 < \left|f(\frac{1}{n})\right| < e^{-n}$ for $n = 2, 3, 4, \ldots$? Justify your answer.

Let $D = \left\{ z \in \mathbb {C} : \left|z\right| < 1\right\}$. Find all holomorphic functions $f : D \to \mathbb {C}$ such that $f\! \left(\frac{1}{n} + ie^{-n}\right)$ is real for all integers $n \geq 2$.

Find a one-to-one holomorphic map from the unit disk $\left\{ \left|z\right| < 1\right\}$ onto the slit disk $\left\{ \left|w\right| < 1\right\} - [0,1)$.

Map the disk $\left\{ \left|z\right| < 1\right\}$ with slits along the segments $[a,1], [-1,-b]$ ($0 < a < 1$, $0 < b < 1$) conformally onto the full disk $\left\{ \left|w\right| < 1\right\}$ by means of a function $w = f(z)$ with $f(0) = 0$, $f'(0) > 0$. Compute $f'(0)$ and the lengths of the arcs corresponding to the slits.

Let $0 < \varepsilon < \pi$, let $\gamma_\varepsilon$ denote the arc $\left\{ e^{it} : \varepsilon \leq t \leq 2\pi -\varepsilon \right\}$ and let $\Omega_\varepsilon$ be the complement of $\gamma_\varepsilon$ in the Riemann sphere. If $f$ is the conformal map of the unit disk onto $\Omega_\varepsilon$, $f(0) = 0$, $f'(0) > 0$, describe the part of the unit disc that $f$ maps onto $\left\{ \left|z\right| > 1\right\}$.

Suppose that $w = f(z)$ maps a simply connected region $G$ one-to-one and conformally onto a circular disk $D_r$ with center $w = 0$, radius $r$, such that $f(a) = 0$ and $\left|f'(a)\right| = 1$ for some point $a \in G$.

Let $T(z) = \dfrac {az+b}{cz+d}$ be a Mobius transformation.

If $f : D(0,1) = \left\{ \left|z\right| < 1\right\} \to \mathbb {C}$ is an analytic function which satisfies $f(0) = 0$, and if

prove that

Let $\Omega = \left\{ z \in \mathbb {C}; -1 < \im z < 1\right\}$, and let $\mathcal F$ be the family of all analytic functions $f : \Omega \to \mathbb {C}$ such that $\left|f\right| < 1$ on $\Omega$ and $f(0) = 0$. Find

Let $f$ be an analytic function on $D = \left\{ z; \left|z\right| < 1\right\}$ such that $f(0) = 1$, and suppose that $\left|1+f(z)\right| \leq 1 + \left|f(z)\right|$ whenever $\left|z\right| < 1$. Prove that $\left|f'(0)\right| \leq 4$.

Let $P$ be the set of holomorphic functions $f$ on the open unit disc so that (i) Both the real and imaginary parts of $f(z)$ are positive for $\left|z\right| < 1$, (ii) $f(0) = 1+i$. Let $E = \left\{ f(\frac12) : f \in P\right\}$. Describe $E$ explicitly.

Let $\Omega = \left\{ z : \left|z-4\right| < 4\right\}$. If $\mathcal F$ is the family of all analytic functions on $\Omega$ such that $\left|f\right| < 1$ in $\Omega$ and $\lim_{w \to \infty } f(w) = 0$, find $\sup_{f \in \mathcal F} \left|f(8)\right|$. Your answer should be an explicit number, and you should prove your assertion.

Let $D$ be the upper half plane and let $f \neq \mathrm{id}$ be a conformal map of $D$ onto itself such that $f \circ f = \mathrm{id}$. Prove that $f$ has a unique fixed point inside $D$.

Let $\Omega$ be a convex, open subset of $\mathbb {C}$ and let $f : \Omega \to \mathbb {C}$ be an analytic function satisfying $\mathrm{Re}\left(f'(z)\right) > 0$, $z \in \Omega$. Prove that $f$ is one-to-one in $\Omega$ (i.e., $f$ is injective).

Show that if the polynomial $P(z) = a_n z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$, $n > 1$, is one-to-one in the unit disk $\left|z\right| < 1$ and $a_1 = 1$, then $\left|na_n\right| \leq 1$.

Let $P(z)$ be a polynomial on the complex plane, not identically zero; let $H = \left\{ z : \re z > 0\right\}$.

Let $f(z)$ be a Laurent series centered at $0$, convergent in $\mathbb {C} \setminus \left\{ 0\right\}$, with residue $b$ at $z=0$.

Assume $f$ is analytic in a neighborhood of $\closure D$, $f$ maps $D$ into $D$, and $f$ maps $\partial D$ into $\partial D$, where $D = \left\{ z : \left|z\right| < 1\right\}$.

If $f(z)$ is a univalent (1-1 analytic) function with domain the unit disc such that $f(z) = z\sum_{n=2}^{\infty } a_n z^n$, then prove that

is a univalent function in the unit disc.

Consider the unit disk $D$ and an analytic function $f : D \to D$. If $f(0) = a \neq 0$, show that $f$ does not have a zero in the disk $\left\{ z : \left|z\right| < \left|a\right|\right\}$.

Let $f(z) = \dfrac {z}{(1+z)^2}$. Let $V = \left\{ z \in \mathbb {C} : 0 < \left|z\right| < \frac12, \frac{3}{4}\pi < \arg z < \frac54\pi \right\}$.

Evaluate the integral

Evaluate

where $\gamma$ is the positively oriented circle $\left\{ \left|z\right| = 1\right\}$.

For what value of $a$ is the function

single-valued?

Define

What is the largest possible $P$ so that $h(z)$ is analytic for $\left|z\right| < P$?

Let $f(z)$ be analytic in $S = \left\{ z \in \mathbb {C}; \left|z\right| < 2\right\}$. Show that

Suppose that the real-valued function $u$ is harmonic in the disk $\left\{ \left|z\right| < 2\right\}$, $v$ is its harmonic conjugate and $u(0) = v(0) = 0$. Show that

where $\gamma (t) = e^{2\pi i t}$, $t \in [0,1]$.

Let $f$ be an analytic function on an open set containing $\overline{D(0,1)} = \left\{ z; \left|z\right| \leq 1\right\}$.

If $f$ is analytic in the unit disk and its derivative satisfies

show that the coefficients in the expansion

satisfy $\left|a_n\right| \leq e$ for $n \geq 1$, where $e$ is the base of natural logarithms.

Let $f = u+iv$ be an entire function.

Let $f$ be an entire function that satisfies $\left|\mathrm{Re}\left(f(z)\right)\right| \leq \left|z\right|^n$ for all $z$, where $n$ is a positive integer. Show that $f$ is a polynomial of degree at most $n$.

Compute the double integral

where $D$ is the disk given by $\left\{ z = x+iy \in \mathbb {C} : x^2+y^2 < 1\right\}$.

Let

be analytic in $D = \left\{ \left|z\right| < 1\right\}$ and assume that the integral

is finite.

Let $f$ be analytic in $\left\{ 0 < \left|z\right| < 1\right\}$ and in $L^2$ with respect to planar Lebesgue measure. Is $0$ a removable singularity? Proof or counterexample.

Evaluate the integral

considering separately the cases $\rho < 1$, $\rho > 1$.

Evaluate

by the method of residues.

Consider the function