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Homework 14: Complex Analysis

Due May 22nd Problem 1 Let $latex a\in\mathbb C$ and $latex f,g\in\mathscr O_a$. Then $latex f = g$ if and only if $latex f^{(k)}(a) = g^{(k)}(a)$ for all $latex k\in\mathbb N$. Problem 2 Let $latex \gamma_0, \gamma_1$ paths from $latex a$ to $latex b$ in $latex V$. Then $latex \gamma_0$ and $latex \gamma_1$ are path-homotopic in $latex V$ if and only if $latex \gamma_0-\gamma_1$ is homotopic to a constant in $latex V$, where $latex \gamma_0-\gamma_1:[0,1]\to\mathbb C$ is defined by $latex (\gamma_0-\gamma_1)(t) = \begin{cases}\gamma_0(2t) & 0\le t\le 1/2\\ \gamma_1(2-2t) & 1/2\le t\le 1. \end{cases}$ Problem 3 Let $latex V\subset W\subset\mathbb C$ open connected, $latex u:W\to\mathbb R$ harmonic and $latex f\in H(V)$ such that $latex \Re f = u$ in $latex V$. Then $latex (f,V)$ admits unrestricted continuation in $latex W$.

Homework 13: Complex Analysis

Due May 15th Problem 1 Let $latex f:[0,1]\to(0,1)$ continuous and $latex V=\{x+iy:x\in(0,1), f(x) < y < 1\}$. Then every point $latex x + if(x), x\in(0,1)$ in $latex \partial V$ is simple. Problem 2 Define $latex f\in H(\mathbb D)$ by $latex f(z) = e^{i/(z-1)^2}$ and $latex \gamma:[0,1]\to\mathbb C$ by $latex \gamma(t) = 1 + (1-t)e^{3\pi i/4}$. $latex \gamma([0,1))\subset\mathbb D$ (draw a picture) $latex \displaystyle \lim_{t\to1} f(\gamma(t))$ exists $latex \displaystyle \lim_{r\to 1} f(r)$ does not exist Find $latex \gamma_1$ such that $latex |f(\gamma_1(t))|\to\infty$ as $latex t\to1$

Homework 12: Complex Analysis

Due May 8th Problem 1 If $latex P(z)$ is a polynomial, then there exists $latex z_0$ such that $latex |z_0|=1$ and $latex |P(z_0) - \dfrac{1}{z_0}| \ge 1$. Problem 2 There exists a sequence $latex P_n(z)$ of polynomials such that $latex P_n(0) =1$ for all $latex n$ and $latex P_n(z)\to 0$ for all $latex z\in\mathbb C$, $latex z\not=0$. Problem 3 Assume Runge's Theorem and prove the following theorem: Let $latex V\subset\mathbb C$ open and $latex \Gamma$ a cycle in $latex V$. If   $latex \displaystyle \int_\Gamma f(z) dz = 0$  for all $latex f\in H(V)$ of the form $latex f(z) = 1/(z-a)$, then  $latex \displaystyle \int_\Gamma f(z) dz = 0$  for all $latex f\in H(V)$. Problem 4 Let $latex V\subset\mathbb C$ be bounded, connected and open. There exists $latex f\in H(V)$ which cannot be extended to a function holomorphic in a strictly larger open set.

Homework 11: Complex Analysis

Problem 1 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex g:\mathbb R\to\mathbb R$ by $latex g(y) = \Re f(iy)$. Then $latex g$ is even. Problem 2 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex h:\mathbb R\to\mathbb R$ by $latex h(y) = \Im f(iy)$. Then $latex h$ is an odd function. Problem 3 Verify the previous problems with the functions $latex \sin z, \cos z, 1 + z^2 + z^3$. Problem 4 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$ and $latex f(iy)$ purely imaginary for $latex y\in\mathbb R$. Then $latex f$ is odd. Problem 5 What can you say if, instead, $latex f(iy)$ is also real valued for $latex y\in\mathbb R$?

Homework 10: Complex Analysis

Due April 24th Problem 1 Let $latex U,V\subset\mathbb C$ open, $latex f:U\to V$ holomorphic and $latex u:V\to\mathbb C$ harmonic. The $latex u\circ f$ is harmonic in $latex U$. Problem 2 Let $latex f\in C(\partial \mathbb D)$ and $latex \psi\in\text{Aut}(\mathbb D)$. Then $latex \mathscr P(f\circ\psi) = \mathscr Pf\circ\psi$. (A sketch of the proof of this result is given in the text --Theorem 10.2.0--; give the details.)  Problem 3 Explain why the following "proof" that there is no continuous function on $latex \bar{\mathbb D}$, holomorphic in $latex \mathbb D$ and equal to $latex f(e^{it}) = e^{-it}$ on the boundary, is wrong: Let $latex u\in C(\bar{\mathbb D})$ be holomorphic in $latex \mathbb D$ with $latex u|_{\partial\mathbb D} = f$. Then $latex v(z) = u(z) - 1/z$ is holomorphic in $latex \mathbb D\setminus\{0\}$, vanishing on the boundary of $latex \mathbb D$. Since the set $latex \partial\mathbb D$ has an accumulation point and $latex \mathbb D\setminu

Homework 9: Complex Analysis

Problem 1 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$. If $latex \mathcal F$ is a normal family, then $latex \mathcal F' = \{f': f\in\mathcal F\}$ is also a normal family. Problem 2 The converse to the result in Problem 1 is false. Find a "small" hypothesis to make the converse true. Problem 3 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$ a normal family. Let $latex f_n\in \mathcal F$ a sequence such that $latex f_n(z)\to f(z)$ for each $latex z\in S\subset V$, where $latex S$ has a limit point in $latex V$ and $latex f\in H(V)$. Then $latex f_n\to f$ in $latex H(V)$. Problem 4 Let $latex V\subset \mathbb C$ be open, $latex M>0$, and  $latex \displaystyle\mathcal F = \Big\{ f\in H(V): \iint_V |f(z)|^2 dxdy \le M \Big\}$. Then $latex \mathcal F$ is a normal family. Problem 5 Let $latex V\subset \mathbb C$ be open and connected and $latex \{f_n\}\subset H(

Homework 8, Complex Analysis

Problem 1 Consider the function on $latex [0,\infty)$ given by $latex \psi(t) = \dfrac{t}{1+t}$. $latex \psi$ is concave and increasing $latex \psi$ is continuous at $latex 0$ For any $latex \varepsilon > 0$ there exists $latex \delta > 0$ such that $latex \psi(t) < \delta$ implies $latex 0\le t < \varepsilon$ Problem 2 Let $latex d_j$ be a sequence of quasimetrics on $latex X$ that separates points, and $latex d$ the metric constructed from the $latex d_j$ as in class. Let, for each positive integer $latex N$, $latex x\in X$ and $latex \varepsilon>0$, $latex B_N(x,\varepsilon) = \{ y\in X: d_j(x,y) < \varepsilon, j = 1, 2, \ldots, N\}.$ Then $latex B_N(x,\varepsilon)$ is open with respect to the metric $latex d$. Problem 3 $latex S\in X$ is totally bounded in $latex (X,d)$ if and only if for any $latex \varepsilon > 0$ and any positive integer $latex N$ there exist $latex x_1, x_2, \ldots, x_m$ such that $latex \displaystyle S \subset \bigc

Homework 7: Complex Analysis

Problem 1 The map $latex \Phi(z) = i\dfrac{1+z}{1-z}$ is a biholomorphic mapping from the unit disk $latex \mathbb D$ onto the upper half-plane $latex \mathbb H$. $latex \Phi$ is called the Cayley transform . Problem 2 Let $latex A=\begin{pmatrix} a & b\\c & d \end{pmatrix}\in SL(2,\mathbb R)$ and $latex \phi_A$ the Möbius transformation $latex \phi_A(z) = \dfrac{az+b}{cz+d}$. Then $latex \phi_A\in\text{Aut}(\mathbb H)$. Problem 3 For $latex a\in\mathbb H$, give an explicit $latex A\in SL(2,\mathbb R)$ so that $latex \phi_A$ takes $latex a$ to $latex i$. Problem 4 If $latex \psi\in\text{Aut}(\mathbb H)$, then there exists $latex A\in SL(2,\mathbb R)$ so that  $latex \psi = \phi_A$. ( Hint:  Consider the matrix $latex A = \begin{pmatrix} \cos\theta & -\sin\theta\\\sin\theta & \cos\theta \end{pmatrix}$ and verify that $latex \Phi^{-1}\circ\phi_A\circ\Phi$ is a rotation in $latex \mathbb D$. Use the previous problem and the discussion in class, with

Homework 6: Complex Analysis

Due March 13th Problem  1 Let $latex z\mapsto P(z)$ be the stereographical projecton. Then $latex P(z)$ and $latex P(w)$ are antipodal points if and only if $latex z\bar w = -1$. Problem 2 Let $latex f(z)$ be an entire function such that $latex \lim_{z\to\infty} f(z) = \infty$. Prove, without any use of the Riemann sphere, that $latex f(z)$ is a polynomial. Problem 3 Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C)$ is equal to the set of nonconstant affine maps. Problem 4 Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C_\infty)$ is equal to the set of all linear-fractional transformations. Problem 5 For a non-singular complex $latex 2\times 2$ matrix $latex A$, let $latex \phi_A$ be the linear-fractional transformation seen in class. Verify directly that $latex \phi_A\circ\phi_B = \phi_{AB}$. For$latex A\in GL_2(\mathbb C)$, let $latex P_A$ be the mappi

Homework 5: Complex Analysis

Due March 6th Problem 1 Suppose $latex f\in H(\mathbb C)$, $latex f(z+1) = - f(z)$ for all $latex z\in\mathbb C$, $latex f(0) = 0$ and $latex |f(z)| \le e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$. Then $latex f(z) = c\sin\pi z$ for some constant $latex c$. ( Hint:  Use Problem 4 of Homework 2.) Problem 2 Suppose $latex f:\mathbb C\to\mathbb C$ is continuous, $latex f(z + 2) = f(z)$ and $latex |f(z)| \le e^{\pi|z|}$ for all $latex z\in\mathbb C$. Then there exists a constant $latex c$ such that $latex |f(z)| \le c e^{\pi|\Im z|}$ for all $latex z\in\mathbb C$. Problem 3 Find $latex \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ using the series for $latex \cot\pi z$. Problem 4 Prove that Theorem 7.5 in the text follows from Theorem 7.6. Problem 5 Suppose $latex f\in H(V)$, $latex W\subset \mathbb C$ is open, and $latex h:W\to V$ satisfies $latex f(h(z)) = z$ for all $latex z\in W$. Give an example where it is not true that $latex h\in H(W)$. If $latex h$ is

Homework 4: Complex Analysis

Due February 28th Problem 1 For any positive integer $latex n$, the polynomial $latex z^n(z-2)-1$ has $latex n$ roots in the disk $latex \mathbb D = D(0,1)$. Problem 2 Suppose $latex f:\overline{\mathbb D}\to\overline{\mathbb D}$ is continuous and holomorphic in $latex \mathbb D$. Then $latex f$ has a fixed point in $latex \overline{\mathbb D}$. Problem 3 Prove Hurwitz's Theorem: If $latex V$ is an open set, $latex f_n\in H(V)$ is a sequence such that $latex f_n\to f$ uniformly on compact subsets of $latex V$, $latex \overline{D(z,r)}\subset V$, and $latex f$ has no zeroes on $latex \partial D(z,r)$, then there exists $latex N$ such that, for all $latex n\ge N$, $latex f_n$ and $latex f$ have the same number of zeroes in $latex D(z,r)$. Problem 4 The series $latex \displaystyle \sum_{n=0}^\infty \frac{1}{z-n}$ diverges for all $latex z\in\mathbb C\setminus\mathbb Z$. The series $latex \displaystyle \sum_{n=0}^\infty \Big( \frac{1}{z-n} + \frac{1}{z+n} \Big)$ conv

Homework 3: Complex Analysis

Due February 21st Problem 1 Verify explicitly that $latex \text{Ind}(\partial D(z_0,r),z_0) = 1$ for all $latex z_0\in\mathbb C$ and $latex r>0$. For $latex z_0\in\mathbb C$ and $latex r>0$, $latex \text{Ind}(\partial D(z_0,r),z) = \begin{cases} 1 & |z-z_0| < r\\ 0 & |z-z_0| > r. \end{cases}$ Problem 2 If $latex f,g$ are holomorphic near $latex z_0$ and $latex f$ has a simple zero at $latex z_0$, find an expression for the residue of $latex g/f$ at $latex z_0$. If $latex f$ has a simple pole at $latex z_0$ and $latex g$ is holomorphic near $latex z_0$, then $latex \text{Res}(fg,z_0) = g(z_0)\text{Res}(f,z_0)$. If $latex f$ is holomorphic near $latex z_0$ and $latex g(z) = f(z)/(z-z_0)^n$, then $latex \text{Res}(g,z_0) = \dfrac{f^{(n-1)}(z_0)}{(n-1)!}$. Problem 3 Use the residue theorem to show that $latex \displaystyle \int_{-\infty}^\infty \frac{dx}{1+x^2} = \pi$. ( Hint:  For $latex R>0$, consider the upper semicircle $latex \gamma$

Homework 2: Complex Analysis

Due February 14th Problem 1 Explain why the function $latex f(z) = z^2\sin(1/z)$ for $latex z\not=0$, $latex f(0) = 0$, is not a counterexample to the result that, if a function is differentiable in a neighborhood of a point, then it is infinitely differentiable at that point. Problem 2 Verify explicitly that $latex \displaystyle \int_{\partial T}\frac{1}{z}dz = 2\pi i$ for the triangle $latex T = [1, -1+i, -1-i]$. Problem 3 Prove the mean value property : If $latex V\subset\mathbb C$ is open, $latex f\in H(V)$ and $latex \overline{D(z_0,r)}\subset V$, then $latex \displaystyle f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + r e^{it}) dt$. Problem 4 Let $latex f\in H(\mathbb C)$. If $latex |f(z)| \le e^{\Re z}$ for all $latex z$, then there exists a constant $latex c$ such that $latex f(z) = c e^z$. If there is $latex n\in\mathbb Z_+$ such that $latex |f(z)| \le (1 + |z|)^n$ for all $latex z$, then $latex f$ is a polynomial. If $latex f(n) = 0$ for all $latex n\i

Homework 1: Complex Analysis

Due February 7th Problem 1 If $latex f$ is differentiable at $latex z$, then $latex f$ is continuous at $latex z$. Problem 2 Let $latex T:\mathbb C \to \mathbb C$ be $latex \mathbb R$-linear. $latex T$ is $latex \mathbb C$-linear if and only if $latex T(iz) = iTz$ for all $latex z\in\mathbb C$. If $latex T$ has matrix $latex \begin{pmatrix}a & b\\c & d\end{pmatrix}$, then $latex T$ is $latex \mathbb C$-linear if and only if $latex a = d$ and $latex b = -c$. Problem 3 Define $latex f:\mathbb C\to\mathbb C$ by $latex f(x+iy) = \begin{cases} 0 & x=0\\ 0 & y=0\\ 1 & \text{otherwise.}\end{cases}$ Then $latex f$ satisfies the Cauchy-Riemann equations at the origin, but it is not differentiable at the origin. Problem 4 Define $latex f:\mathbb C\to\mathbb C$ by $latex f(z) = \begin{cases} |z|^2\sin(1/|z|) & z\not=0\\ 0 & z=0.\end{cases}$ Then $latex f$ is differentiable at the origin, but the partial derivative $latex \partial_x u$ is