Ian Charlesworth

1. Automorphisms of the disc

   1. Let $a, b \in \D$ be distinct, and suppose that $f, g \in \aut(\D)$ are such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that $f = g$.
   2. Let $a, b \in \D$ (not necessarily distinct), and suppose that $f, g \in \aut(\D)$ are such that $f(a) = g(a)$ and $f'(b) = g'(b)$. Show that $f = g$.

   3. Must every automorphism of $\D$ have a fixed point? That is, given $f \in \aut(\D)$, is there necessarily some $z \in \D$ so that $f(z) = z$?

2. Injective maps and poles

   Suppose that $f : \Omega \to \C$ is injective with a singularity at $s$.
   1. Show that if the singularity is removable, then the holomorphic extension of $f$ across the singularity remains injective. (Hint: the Open Mapping Theorem may be useful here.)
   2. Show that if the singularity is not removable, then it is a pole of order $1$, and that $f$ can have at most one pole.

3. Möbius transformations

   Recall that given \[M = \begin{bmatrix}a&b\\c&d\end{bmatrix} \in \mat\] such that $\det(M) = ad-bc \neq 0$, we define the corresponding Möbius transformation \begin{align*} f_M : \C_\infty &\longrightarrow \C_\infty \\ z &\longmapsto \frac{az+b}{cz+d}. \end{align*} Recall also that $\det(M)\neq 0$ if and only if $M$ is invertible. Throughout this entire problem, all matrices are assumed to be invertible.
   1. Verify that for $A, B \in \mat$, \[f_A\circ f_B = f_{AB}.\]
   2. Show that for any $M \in \mat$, $f_M$ is a bijection.

   Since $f_M$ is holomorphic on its domain and its singularity is a pole, this tells us $f_M \in \aut(\C_\infty)$.

   3. Describe all $M$ for which $f_M(0) = 0$ and $f_M(\infty) = \infty$.
   4. Suppose that $\varphi \in \aut(\C_\infty)$ is such that $\varphi(0) = 0$ and $\varphi(\infty) = \infty$. Show that for some $\lambda \in \C^\times$, $\varphi(z) = \lambda z$. (Hint: show $\varphi(1/z)$ has a simple pole at zero; remove it.)
   5. Conclude that $\aut(\C_\infty) = \set{f_M \mid M \in \mat}$.

   We can conclude that any automorphism of $\C_\infty$ is completely determined by its pole, its zero, and a scale. Alternatively, given three input points and three output points, there is precisely one automorphism of $\C_\infty$ which takes each input point to its corresponding output.

1. Functions with non-zero derivative

   1. Give an example of open sets $U, V \subseteq \C$ and a function $f : U \to V$ so that $f'(z) \neq 0$ for all $z \in U$, $f(U) = V$, but $f$ is not injective.

   2. Suppose $U, V \subseteq \C$ are open, and $f : U \to V$ is holomorphic. We say $f$ is a local bijection if for every $z_0 \in U$ there is $\delta \gt 0$ so that \[f|_{B_\delta(z_0)} : B_\delta(z_0) \longrightarrow f(B_\delta(z_0))\] is a bijection.

      Prove that $f$ is a local bijection if and only if $f'(z) \neq 0$ for all $z \in U$.

2. Boundary values on the upper half-plane

   Suppose $f : \H \to \C$ is holomorphic and extends continuously to $\H \cup \R \cup \set{\infty}$. Show that if for all $x$ in some interval $[a, b] \subseteq \R$ (with $a \lt b$) we have $f(x) = 0$ then $f \equiv 0$. (Hint: you may find a result from a previous assignment to be useful here.)

   As a bonus, determine if the same holds if we merely assume that $f$ extends continuously to $\H \cup [a, b]$, not necessarily all of $\H\cup\R\cup\set{\infty}$.

3. Simple connectivity

   Suppose that $U, V \subseteq \C$ are conformally equivalent. Prove that $U$ is simply connected if and only if $V$ is.

4. The plane, the disc, and the sphere are distinct

   Recall that a function $f$ defined on $\C_\infty$ is holomorphic if it is holomorphic at every point in $\C$ and has a removable singularity at $\infty$. Prove that no two of $\C$, $\C_\infty$, and $\D$ are conformally equivalent.

1. Laurent series

   Recall from Assignment 7 that if $0 \lt r \lt R \lt \infty$ and $f$ is holomorphic on (an open region containing) $\overline{A_{r, R}(0)}$ then for $z \in A_{r, R}(0)$ we have \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}.\]
   1. Use the generalized Cauchy Integral Formula to produce a much simpler proof of this fact.
   2. Let $r \gt 0$ and $C_r = \partial B_r(0)$ the circle centred at $0$ of radius $r$. Suppose that $f$ is holomorphic on some open set containing $C_r$. Show that \[g_r : z \longmapsto \frac1{2\pi i}\int_{C_r} \frac{f(w)}{w-z}\,dw\] is holomorphic on $\C \setminus C_r$, with a removable singularity at $\infty$ so that $g(\infty) = 0$.
   3. The right-hand side of the identity \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}\] makes sense for any $z \in \C \setminus (C_r \cup C_R)$. Does it agree with $f$ outside of $\overline{A_{r, R}(0)}$?
   4. Use the power series representation of $g_r(1/z)$ to show that there are coefficients $(a_n)_{n\lt0}$ so that for $|z|\gt r$, \[g_r(z) = \sum_{n=-\infty}^{-1} a_nz^n,\] and the convergence is uniform on compact subsets of $\C\setminus\overline{B_r(0)}$.
   5. Conclude that if $f$ is holomorphic on an open set containing $\overline{A_{r, R}(0)}$ then there are coefficients $(a_n)_{n\in\Z}$ so that \[f(z) = \sum_{n=-\infty}^{\infty} a_nz^n\] uniformly on compact subsets of $A_{r, R}(0)$.
   6. Suppose now that $f$ is holomorphic on an open set containing $\overline{A_{r, R}(z_0)}$. Show that there are coefficients $(a_n)_{n\in\Z}$ so that \[f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n\] uniformly on compact subsets of $A_{r, R}(z_0)$.
   7. For $|z| \neq 1$, compute \[\int_{C_1} \frac{1}{w(w-z)}\,dw.\]

2. Obstructions to primitives

   Suppose $\Omega_1 \subseteq \C$ is open and simply connected, $s_1, \ldots, s_n \in \Omega_1$ are distinct, $S = \set{s_1, \ldots, s_n}$, and $\Omega = \Omega_1 \setminus S$.
   1. Let $f : \Omega \to \C$ be holomorphic. Show that $f$ admits a primitive if and only if for every $s \in S$, $\res_s(f) = 0$.

   2. Show that there are holomorphic functions $\varphi_1, \ldots, \varphi_n : \Omega \to \C$ so that for any $f : \Omega \to \C$ holomorphic, there are coefficients $\alpha_1, \ldots, \alpha_n \in \C$ so that \[f + \alpha_1\varphi_1 + \alpha_2\varphi_2 + \ldots + \alpha_n\varphi_n\] admits a primitive.

      (What you have proven is that if $\mathcal{H}_\Omega$ is the vector space of functions holomorphic on $\Omega$ and $\mathcal{H}^0_{\Omega}$ is the vector space of functions on $\Omega$ which admit primitives, then \[\dim_\C\paren{\mathcal H_\Omega / \mathcal H^0_\Omega} = n\text{.)}\]

1. Maximum modulus problems

   1. Suppose that $f, g : \overline{B_1(0)} \to \C$ are holomorphic on $B_1(0)$ and continuous, with $f(e^{it}) = g(e^{it})$ for all $t \in \R$. Show that $f \equiv g$.

   2. Suppose that $f : \overline{B_1(0)} \to \C$ is holomorphic on $B_1(0)$ and continuous, and that for some $\theta \gt 0$, $f(e^{it}) = 0$ for all $t \in [0, \theta]$. Show that $f \equiv 0$. (Hint: consider a product of functions of the form $z \mapsto f(\zeta z)$ with $|\zeta| = 1$.)

      Notice that this allows us to strengthen Part A: the result there still holds even if $f$ and $g$ agree merely on some arc of the circle.

2. Primitives on domains without simple connectivity

   Suppose that $\Omega \subseteq \C$ is a domain, but not necessarily simply connected, and that $f : \Omega \to \C$ is holomorphic. Show that $f$ admits a primitive on $\Omega$ if and only if for every closed curve $\gamma$ in $\Omega$, \[\int_\gamma f(w)\,dw = 0.\]

3. Some branches of logarithm

   1. Suppose that $\gamma$ is a curve in $\C$. Show that there is a unique unbounded connected component of $\C\setminus\gamma$. (Recall that for open sets $U \subseteq \Omega \subseteq \C$, $U$ is a connected component of $\Omega$ if and only if $U$ is non-empty and connected, and $U^c\cap \Omega$ is open.)

      It turns out to also be true that $\C\setminus\gamma$ has a unique bounded component whenever $\gamma$ is simple and closed, but this is much harder to show.

   2. Let $\alpha : [0, \infty) \to \C$ be continuous and injective (i.e., one-to-one), with $\alpha(0) = 0$ and $\lim_{t\to\infty}\abs{\alpha(t)} = \infty$. Set $V_\alpha = \C\setminus\alpha([0, \infty)).$

      Show that for any closed curve $\gamma$ in $V_\alpha$, \[\int_\gamma \frac{dz}z = 0.\] Conclude that $V_\alpha$ admits a branch of logarithm. (Hint: You may wish to apply homotopy invariance on $\C^\times$ rather than $V_\alpha$.)

      It turns out showing that $V_\alpha$ is connected is trickier than I realized. Show instead that the unbounded component $C$ of $V_\alpha$ admits a function $L : C \to \C$ such that $e^{L(z)} = z$ for all $z \in C$. As a bonus, show that $V_\alpha$ is connected and simply connected.

   3. Let $\alpha(t) = te^{it}$. For $z \in V_\alpha$, denote by $\theta_z$ the unique element of $(0, 2\pi)$ for which $|z|e^{i|z|}=ze^{i\theta_z}$. Compute, in terms of $\theta_z$, an explicit formula for the branch of logarithm $\log_{V_\alpha}$ (i.e., the one defined on $V_\alpha$ which takes the value $0$ at $1$.)

      Draw a sketch of the region $\log_{V_\alpha}(V_\alpha)$, but do not submit it.

4. Some exponentiation

   In the following question, we will work always with the principal branch of $z^\alpha$: the one defined for $z \in \C\setminus(-\infty, 0]$ corresponding to the logarithm taking value $0$ at $1$.
   1. Compute $(-i)^{1/2}$ and $5^{i+1}$.
   2. Let $\alpha \gt 0$. Determine all points $t_0 \in (-\infty, 0]$ for which \[\lim_{\substack{z \to t_0 \\ z \in \C\setminus(-\infty, 0]}} z^\alpha\] exists.
   3. Determine the range of $z^i$. (I recommend, but do not require, that you also produce a diagram indicating the value $z^i$ takes on a typical ray from the origin and on a typical circle about $0$.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Another zero counting problem

  Let $\lambda \gt 1$, and define $f_\lambda(z) = \lambda - z - e^{-z}$. Prove that $f_\lambda$ has a unique zero in the right half-plane $H = \set{z \in \C \mid \Re(z) \gt 0}$. Moreover, if $x_\lambda$ is this zero, show that $x_\lambda \in \R$ and $\lim_{\lambda\to1^+}x_\lambda = 0$.

1. More residues

   Suppose that $f : D_r'(z_0) \to \C$ is holomorphic, with a simple pole at $z_0$. Let $g : B_r(z_0)$ be holomorphic with $g(z)f(z) = 1$ for $z$ in the domain of $f$. Show that \[\res_{z_0}(f) = \frac{1}{g'(z_0)}.\] (Notice, also, that this approach will fail if $f$ has a pole of order more than one at $z_0$.)

2. Some integrals

   1. Let $p, q$ be polynomials with complex coefficients, so that $q$ admits no roots in $\R$ and $\deg q \geq \deg p + 2$. Show that \[\int_{-\infty}^\infty \frac{p(x)}{q(x)}\,dx = 2\pi i \sum_{j=1}^k \res_{z_j}\paren{\frac pq},\] where $z_1, \ldots, z_k$ are the zeros of $q$ with $\Im(z_k) \gt 0$.
   2. Compute, for $n \in \N$, $n \geq 2$, the integral \[\int_0^\infty \frac{dx}{x^n+1}.\] (Hint: use the "wedge" $[0, R]\,. \sigma_R\,. [R\zeta_{n}, 0]$ where $\zeta_n = e^{2\pi i / n}$ and $\sigma_R(t) = Re^{it}$ for $t \in [0, 2\pi/n]$.)
   3. Compute the integral \[\int_{-\infty}^\infty \frac{2x^2+1}{(x^4+1)(x^2+1)}\,dx.\]

3. Perturbations of zeros

   Suppose that $r \gt 1$, and $f, g : B_{r}(0) \to \C$ are holomorphic. Suppose further that $f$ has a simple zero at $0$, and vanishes nowhere else in $\overline{B_1(0)}$. Let $f_\epsilon = f(z) + \epsilon g(z)$.
   1. Show that if $\epsilon$ is sufficiently small, then $f_\epsilon$ has a unique zero in $B_1(0)$.
   2. Let $z_\epsilon$ be the unique zero of $f_\epsilon$ from part A. Show that $\epsilon \mapsto z_\epsilon$ is continuous.

   (A consequence of this is that the zeros of a polynomial are continuous functions of its coordinates. Reflect on how to state this precisely.)

4. A counting problem

   Determine the number of zeros of $p(z) = 1 + 6z^3 + 3z^{10}+z^{11}$ in $A_{1, 2}(0)$. (Hint: determine the number of zeros in $B_1(0)$ and in $B_2(0)$. This can be done using either the Argument Principle or Rouché's Theorem and some estimates.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Further integrals

  1. Show that \[\int_{-\infty}^\infty \frac{dx}{(1+x^2)^{n+1}} = \frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\pi = \frac{(2n)!}{2^{2n}(n!)^2}\pi.\]
  2. Show that \[\int_0^1 \log(\sin \pi x)\,dx = -\log2.\] (Hint: use a rectangular contour with base $[0, 1] .[1, 1+iR].[1+iR,iR].[iR,0]$ and let $R \to\infty$.)

At the beginning of lecture on Tuesday, we will have the following definition:

1. Homotopy is an equivalence relation

   Let $\Omega \subseteq \C$. Given closed curves $\gamma_0, \gamma_1$ in $\Omega$, let us write $\gamma_0 \sim_\Omega \gamma_1$ if they are homotopic in $\Omega$. Prove that $\sim_\Omega$ is an equivalence relation. That is, for any $\gamma_0, \gamma_1, \gamma_2$ closed curves in $\Omega$: $\gamma_0 \sim_\Omega \gamma_0$; $\gamma_0\sim_\Omega \gamma_1$ if and only if $\gamma_1\sim_\Omega\gamma_0$; and if $\gamma_0 \sim_\Omega \gamma_1$ and $\gamma_1\sim_\Omega\gamma_2$ then $\gamma_0\sim_\Omega\gamma_2$.

   (In fact, homotopy of (not necessarily closed) curves is also an equivalence relation, but showing this is essentially the same argument.)

2. Invariance of integrals along homotopic closed curves

   Suppose that Ω is open, that $\gamma_0, \gamma_1$ are closed curves homotopic in $\Omega$, and $h : [0, 1]\times [0, 1] \to \Omega$ is a homotopy between them. Let $\sigma$ be the curve parameterized by \begin{align*} [0, 1] &\longrightarrow \Omega \\ s & \longmapsto h(s, 0) = h(s, 1). \end{align*}
   1. Show that $\gamma_0$ and $\sigma\,.\gamma_1\,.\sigma^-$ are homotopic as curves (rather than as closed curves). In particular, show that there is a homotopy between them which fixes the endpoints of the curves in question.
   2. Conclude that if $\gamma_0, \gamma_1$ are homotopic closed curves in $\Omega$ and $f : \Omega \to \C$ is holomorphic, then \[\int_{\gamma_0}f(z)\,dz = \int_{\gamma_1}f(z)\,dz.\]

3. Homotopies in starlike regions

   Suppose that $\Omega \subseteq \C$ is open and starlike about $z$. Let $\gamma$ be a path in $\Omega$ from $w_0$ to $w_1$. Show that $\gamma$ is homotopic to $[w_0, z]\,.[z, w_1]$. Conclude that any two paths in $\Omega$ from $w_0$ to $w_1$ are homotopic to each other.

4. Annuli

   Suppose $0 \leq r \lt R \leq \infty$ and $z_0 \in \C$. We denote by $A_{r, R}(z_0)$ the annulus of inner radius $r$ and outer radius $R$ centred at $z_0$: \[A_{r, R}(z_0) := \set{z \in \C \mid r \lt \abs{z-z_0} \lt R}.\]

   Suppose that $\Omega \subseteq \C$ is open, $0 \lt r \lt R \lt \infty$, $z_0 \in \C$, and $\overline{A_{r, R}(z_0)} \subset \Omega$.

   1. Show that $\partial B_{r}(z_0)$ and $\partial B_R(z_0)$ are homotopic in $\Omega$.
   2. Suppose that $f : \Omega \to \C$ is holomorphic. Show that for $z \in A_{r, R}$, we have \[f(z) = \frac1{2\pi i}\paren{\int_{\partial B_R(z_0)} \frac{f(w)}{w-z}\,dw - \int_{\partial B_r(z_0)} \frac{f(w)}{w-z}\,dw}.\]

1. Essential singularitites

   Suppose that $\Omega \subset \C$ is open, and $f : \Omega \to \C$ is holomorphic with an essential singularity at $z_0$.
   1. Suppose $g : \Omega \cup\set{z_0} \to \C$ is holomorphic with a zero of order $N$ at $z_0$. Classify the singularity of $fg$ at $z_0$.
   2. Suppose $g : \Omega \to \C$ has a pole at $z_0$. Classify the singularity of $fg$ at $z_0$.

2. Dealing with an essential singularity

   Let $\Omega = \C\setminus\set0$, and consider the function $f : z \mapsto \exp\paren{\frac1z}$ which is holomorphic on $\Omega$.
   1. Show that there is are $a_0, a_{-1}, a_{-2}, \ldots, \in \C$ so that the series \[\sum_{n=-\infty}^0 a_nz^n\] converges uniformly to $f$ on compact subsets of $\Omega$.
   2. Find $a_{-1}$, the residue of $f$ at $0$, explicitly.
   3. Compute \[\int_{\partial \color{red}{B_{123}(0)}} f(z)\,dz.\]

3. No poles of order 1/2

   Suppose $f : D'_r(z_0) \to \C$ is holomorphic, and that for some $\epsilon \gt 0$ and $A \in \R$ we have \[\abs{f(z)} \leq A\abs{z-z_0}^{\epsilon - 1}.\] Show that $f$ has a removable singularity at $0$.

4. The Riemann sphere

   Recall that we defined the Riemann sphere as \[\S := \set{(\alpha, \beta, \gamma) \in \R^3 \mid \alpha^2+\beta^2+\gamma^2=1}.\] We identified $\S$ with the "infinitied" complex plane $\C_\infty$ by \begin{align*} S &\longrightarrow \C_\infty\\ (\alpha, \beta, \gamma) &\longmapsto \begin{cases} \infty & \text{ if } \gamma = 1 \\ \frac{\alpha+i\beta}{1-\gamma} & \text{ otherwise.}\end{cases} \end{align*} Here the inverse mapping is given by \[ x+iy \longmapsto \paren{\frac{2x}{x^2+y^2+1}, \frac{2y}{x^2+y^2+1}, 1-\frac{2}{x^2+y^2+1}}. \]
   1. Describe the map $\S \to \S$ which corresponds to $z \mapsto iz$.
   2. Describe the map $\S \to \S$ which corresponds to $z \mapsto \bar z$.
   3. Describe the map $\C_\infty \to \C_\infty$ which corresponds to sending each point $p \in \S$ to its antipode, $-p$.

   (An example of the level of detail desired: the map $z \mapsto z^{-1}$ corresponds to a rotation by angle $\pi$ about the line through $1$ and $-1$ (i.e., through $(1, 0, 0)$ and $(-1, 0, 0)$). Of course, some justification of this claim would be required.)

5. Meromorphisms and poles

   For each of the following functions, find the largest domain on which they are meromorphic and classify their singularities.
   1. \[f(z) = e^z + e^{-\frac1z}\]
   2. \[g(z) = \frac{1}{\sin(1/z)}\]

1. Limits of holomorphic functions

   Use Morera's theorem to prove the following.

   Suppose that $\Omega \subseteq \C$ is open, and $(f_n)_n$ is a sequence of holomorphic functions which converge uniformly to $f$ on compact subsets of $\Omega$. Then $f$ is holomorphic. (You may assume without proof the fact that if $T \subset \Omega$ is a triangle, then there is a convex open set $U$ with $T \subseteq U \subseteq \Omega$. This follows from essentially the same argument as problem 1 on assignment 4.)

2. Bounds on functions

   1. Suppose that $f : \C \to \C$ is entire with the property that for some $C \gt 0$, for all $z \in \C$, \[\abs{f(z)} \leq Ce^{\Re(z)}.\] Prove that there is a constant $c \in \C$ so that $f(z) = ce^z$.
   2. Suppose that $f : \C \to \C$ is entire. Show that $f$ is a polynomial if and only if there exist a constant $C \gt 0$ and some $n \in \N$ such that for all $z \in \C$ \[\abs{f(z)} \leq C(1 + \abs{z})^n.\]

3. Holomorphic extensions of real functions

   Suppose $f : \R \to \R$, and $\Omega \subseteq \C$ is a domain with $\R \subset \Omega$. Prove that $f$ has at most one holomorphic extension to $\Omega$. (That is, show that if there is a holomorphic function $g : \Omega \to \C$ so that $g|_\R = f$, then $g$ is unique.)

4. Removable singularities and a characterization of sine

   1. Suppose that $\Omega \subseteq \C$ is open, $f, g : \Omega \to \C$ are holomorphic, and $z_0 \in \Omega$ is a zero of order $n$ for $f$ and of order $m$ for $g$, for some $1 \leq m \leq n$. Show that $z_0$ is a removable singularity for $\frac fg$. (You may assume without proof that $\frac fg$ is defined on an appropriate punctured neighbourhood of $z_0$, but you might wish to reflect for a moment on why that is the case.)
   2. Suppose that $f : \C \to \C$ is holomorphic, with $f(n) = 0$ for each $n \in \Z$. Show that each singularity of $z \mapsto \frac{f(z)}{\sin(\pi z)}$ is removable.

   3. Suppose that $f : \C \to \C$ is holomorphic and satisfies $f(z+1) = -f(z)$, $f(0) = 0$, and for some $C \gt 0$, $\abs{f(z)} \leq C\exp(\pi\abs{\Im(z)})$. Show that $f(z) = c\sin($$\pi$$z)$ for some $c \in \C$.

      (Correction: there was a missing $\pi$ in the statement of the problem, and $f$ was meant to be holomorphic.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Fives

  Suppose that $\Omega \subseteq \C$ is open and $f : \Omega \to \C$ is holomorphic. Let us say that $z_0 \in \Omega$ is a five of $f$ of order $N$ if $f^{(N)}(z_0) \neq 5$ but for all $n \lt N$, \[f^{(n)}(z_0) = 5.\]

  Suppose that $f : \Omega \to \C$ has a five of order $N$ at $z_0 \in \Omega$. Prove that there is a holomorphic function $h : \Omega \to \C$ with $h^{(N)}(z_0) \neq 5$ so that \[f(z) = \sum_{n=0}^{N-1} \frac{5z^n}{n!} + \frac{(z-z_0)^N}{N!} h(z).\] Moreover, show that if $z_0$ is a five of infinite order (i.e., if $f^{(n)}(z_0) = 5$ for all $n \in \N_0$), then \[f(z) = 5\exp(z-z_0).\]

• A bound on the derivative

  Let us write $\D = B_1(0) = \set{z \in \C\mid \abs{z} \lt 1}$. Suppose that $f : \D \to \C$ is holomorphic. Show that \[2\abs{f'(0)} \leq \sup\set{\abs{f(z) - f(w)} \mid z, w \in \D}.\]

• More bounds

  1. Suppose that $f : \C \to \C$ is such that $\Re(f(z)) \leq M$ for all $z\in \C$ and some constant $M \in \R$. Prove that $f$ is constant.

1. Discs in open sets

   Let $\Omega \subseteq \C$ be open, and suppose $z \in \Omega$, $r \gt 0$ are such that $\partial B_r(z) \subset \Omega$ (that is, the circle of radius $r$ centred at $z$ is contained in $\Omega$). Show that there is some $R \gt r$ so that $B_R(z) \subseteq \Omega$. (Hint: $\partial B_r(z)$ is compact.)

   Correction: the assumption should be that $\overline{B_r(z)} = \partial B_r(z) \cup B_r(z) \subset \Omega$; otherwise there are many counterexamples.

2. Starlike regions

   Recall that a set $V \subseteq \C$ (or $\R^n$) is called starlike about $z_0$ if for any $z \in V$, we have $[z_0, z] \subseteq V$. A set is starlike if it is starlike about some point it contains.

   Recall that we proved the following in class:

   Suppose that $\Omega \subseteq \C$ is non-empty, open, and convex, and $f : \Omega \to \C$ is a continuous function such that for any triangle $T \subset \Omega$, \[\int_{\partial T} f(z)\,dz = 0.\] Then $f$ admits a primitive on $\Omega$.
   Show that the same conclusion is true if we merely assume that $\Omega$ is non-empty, open, and starlike. (It is enough to identify which part(s) of the proof used convexity, and show that they still work; you do not need to repeat parts of the argument that require no update.)

3. Convex sets

   1. Show that for any $z_0 \in \C$ and any $r \gt 0$, the set $B_r(z_0) = \set{z \in \C \mid \abs{z-z_0} \lt r}$ is convex.
   2. Show that a set $V \subseteq \C$ is convex if and only if for every $n \in \N$, $z_1, \ldots, z_n \in V$, and $a_1, \ldots, a_n \in [0, 1]$ with $a_1+\cdots+a_n = 1$, we have \[a_1z_1 + a_2z_2 + \cdots + a_nz_n \in V.\]

4. Radii of convergence

   1. Suppose that $\Omega \subseteq \C$, $f : \Omega \to \C$ is holomorphic, $z_0 \in \Omega$, and $r \gt 0$ is such that $\overline{B_r(z_0)} \subset \Omega$. Recall that on the disc $B_r(z_0)$, $f$ is given by the power series \[f(z) = \frac1{2\pi i} \sum_{n=0}^\infty \paren{\int_{\partial B_r(z_0)} \frac{f(w)}{(w-z_0)^{n+1}}\,dw}(z-z_0)^n.\] Show that radius of convergence of the series is at least \[\sup\set{R \geq 0 \mid \overline{B_R(z_0)} \subset \Omega}.\]
   2. Let $f : \R \to \R$ be given by \[f(t) = \frac1{1+t^2}.\] Show that $f$ is $\R$-analytic, in the sense that for every $t_0 \in \R$ there are coefficients $a_n \in \R$ and some $\delta \gt 0$ so that for all $t \in (t_0-\delta, t_0+\delta)$, \[f(t) = \sum_{n=0}^\infty a_n(t-t_0)^n.\] Calculate, for every $t_0$, the radius of convergence of this series.

5. Some additional integrals for your enjoyment

   Evaluate the following integrals.
   1. \[\int_{\partial B_r(0)} \frac{\sin(z)}z\,dz \quad(r \gt 0)\quad\]
   2. \[\int_{\partial B_2(0)} \frac{\cos(z)}{1+z^2}\,dz\]
   3. \[\int_{\partial B_r(0)} \frac{z^2+1}{z(z^2+4)}\,dz \quad(r \gt 2 \text{ and } 2 \gt r \gt 0)\]
   Feel free to invoke all the techniques we have available, including Cauchy's integral formula and algebraic manipulations such as partial fractions.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• More convex sets

  Show that if $\set{K_\alpha}_\alpha$ is a collection of convex subsets of $\C$, then \[K = \bigcap_\alpha K_\alpha\] remains convex.

1. Complex exponentiation is tricky

   Let $n \in \N$ with $n \gt 1$. Show that there is no continuous function $f : \C \to \C$ so that for all $z \in \C$, \[f(z)^n = z.\] (Hint: suppose $f$ were such a function, and consider the continuous function $[0, 2\pi] \to \T$ given by $t \mapsto f(\exp(it))$.)

2. Reversals

   Let $\gamma$ be a smooth curve in $\C$, and $f$ a continuous function on $\gamma \subset \C$. (Here we are identifying $\gamma$ with the set of points in the image of any of its parameterizations.) With $\gamma^-$ the reversal of $\gamma$, prove that \[\int_{\gamma^-} f(z)\,dz = -\int_\gamma f(z)\,dz.\]

3. Variety of primitives

   Suppose that $\Omega$ is a domain, and $f : \Omega \to \C$ is continuous. Show that any two primitives of $f$ differ by a constant.

4. Weierstrass's Theorem

   Recall that Weierstrass's theorem states that if $K \subseteq \R$ is compact and $f : K \to \R$ is continuous, then there is a sequence of polynomials (with real coefficients) $(p_n)_n$ which converges uniformly to $f$.

   Is it true that if $K \subseteq\C$ is compact and $f : K \to \C$ is continuous, then there is a sequence of polynomials (with complex coefficients) $(p_n)_n$ which converges uniformly to $f$? (Hint: try integrating polynomials.)

5. Some explicit integrals

   For the purposes of this question, given $a_1, \ldots, a_n \in \C$, let us denote by $[a_1, \ldots, a_n]$ the piecewise smooth curve obtained by concatenating the individual curves $[a_1, a_2]\, . [a_2, a_3]\, . \cdots . [a_{n-1}, a_n]$.

   Compute $\int_\gamma\frac{dz}{z}$, where $\gamma$ is each of the following:

   1. $\sq{1, -1+i, -1-i, 1}$
   2. $\sq{i, -1, -i, 1, i}$
   3. $\sq{1+\frac i2, -1-i, -1+i, 1-\frac i2, -2, 1+\frac i2}$

   You may use $\R$ calculus freely, including its applications to functions such as $\ln$ and $\arctan$ on their real domains. You will save yourself a headache by first devising a formula for $\int_{[a,b]}\frac{dz}z$, provided $0 \notin [a, b]$. Answer using exact values; depending on what approach you take, this may require you to do some bashing of trigonometric functions. If you find yourself trying to compute the sum of several arctangents, one possible route forward is compute both the $\sin$ and $\cos$ of the entire sum, using sum-of-angles formulae and the fact that for $x \in \R$, $\sin(\arctan(x)) = \frac{x}{\sqrt{x^2+1}}$ and $\cos(\arctan(x)) = \frac{1}{\sqrt{x^2+1}}$.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• More integrals

  1. Suppose that $T$ is a triangle and $z_0 \in T \setminus \partial T$. Show that \[\int_{\partial T} \frac{dz}{z-z_0} = 2\pi i.\] Conclude that the orientation on $\partial T$ can be defined as the direction which causes the integral to have this value rather than $-2\pi i$.
  2. Suppose that $\gamma$ is a curve parameterized by $w(t) = z_0 + R\exp(it)$ for $t \in [0, 2\pi]$, some fixed $z_0 \in \C$, and $0 \lt R \lt \abs{z_0}$ (so $\gamma$ is a circle not surrounding $0$). Let $n \in \Z$. Compute \[\int_\gamma z^n\,dz.\]

• The Fundamental Theorem of $\C$alculus

  Suppose that $a \lt b$ are real numbers, and $F : [a, b] \to \C$ is a smooth parameterized curve. Verify that \[\int_a^b F'(t)\,dt = F(b) - F(a).\] (This is the step we glossed over in class on February 6^th. It mostly requires checking how the real and imaginary parts of the derivative of $F$ relate to the derivatives of the real and imaginary parts of $F$.)

1. Products of power series

   1. Let $f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n$ and $g(z) = \sum_{n=0}^\infty d_n(z-z_0)^n$ be power series with radii of convergence $R$ and $S$ respectively. Find a power series expansion for $fg$ centred at $z_0$, and show that its radius of convergence is at least $\min(R, S)$.
   2. Prove that for $w, z \in \C$, we have $e^{w+z} = e^we^z$. Use this and the fact that $e^{iz} = \cos(z) + i\sin(z)$ to deduce the formulae \[\cos(w+z) = \cos(w)\cos(z) - \sin(w)\sin(z) \qquad\text{and}\qquad \sin(w+z) = \cos(w)\sin(z) + \sin(w)\cos(z).\]

2. Directional derivatives

   Suppose $\Omega \subseteq \C$ is open, $f : \Omega \to \C$, $z\in \Omega$, and $\zeta \in \T \color{red}{ := \set{z \in \C : \abs{z} = 1}}$. The directional derivative of $f$ is given by \[D_\zeta f(z) := \lim_{\substack{t\to0\\t\in\R}} \frac{f(z+t\zeta) - f(z)}t,\] provided the limit exists.

   1. Show that if $f$ is holomorphic at $z$ then $D_\zeta f(z) = \zeta f'(z)$ for each $\zeta \in \T$.

   (Remark: Notice that if $f'(z) \neq 0$, then for $\zeta, \xi \in \T$ we can interpret the ratio $D_\zeta f(z) / D_\xi f(z)$ as the angle between the curves $t\mapsto f(z+t\zeta)$ and $t \mapsto f(z+t\xi)$; but then this is just $\zeta\bar\xi$, i.e., the angle between the lines $t \mapsto z+t\zeta$ and $t\mapsto z+t\xi$. For this reason, we say that $f$ "preserves angles" or "is conformal" at $z$.)

   2. Use (a) to produce another proof that if $f$ is holomorphic then it satisifies the Cauchy-Riemann equations.
   3. Prove the following statement, or provide an explicit counterexample: if there is some $a \in \C$ so that for each $\zeta\in\T$ we have $D_\zeta f(z) = a\zeta$, then $f$ is holomorphic at $z$ and $f'(z) = a$.

3. The Ratio Test

   Suppose that $(a_n)_n$ is a sequence in $\C \setminus\set0$ such that \[\limni\abs{\frac{a_{n+1}}{a_n}} = L.\] Show that \[\limni \abs{a_n}^{1/n} = L.\] Note that this provides another method of computing the radii of convergence of some power series.

4. A condition for uniform convergence

   Suppose that $K$ is a compact set and $(f_n)_n$ is a sequence of functions from $K$ to $\C$. Show that $(f_n)_n$ converges uniformly on $K$ if and only if it is uniformly Cauchy, i.e, for every $\epsilon \gt 0$ there is $N \in \N$ so that if $n, m \gt N$ we have $\norm{f_n - f_m}_K \lt \epsilon$.

   (If $(f_n)_n$ is uniformly Cauchy, then for each $z \in K$ the sequence $(f_n(z))_n$ is Cauchy because $\abs{f_n(z) - f_m(z)} \leq \norm{f_n-f_m}_K$. Therefore $(f_n(z))_n$ converges, and so $(f_n)_n$ converges pointwise to a function $f : z \mapsto \limni f_n(z)$. Your goal is to show that this convergence is uniform.)

5. Behaviour at the radius of convergence is delicate

   Devise a power series $f(z) = \sum_{n=0}^\infty a_nz^n$ with radius of convergence 1 which converges uniformly on $\overline{B_1(0)} = \set{z \in \C : \abs{z} \leq 1}$, but the power series describing $f''(z)$ converges at no point of $\T$.

   (If you are stuck, click for a suggestion of some coefficients which will work.)

   Try $a_n = \frac1{n^2}$ for $n \gt 0$. Of course, you still need to prove that these have the desired property.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• More products of power series

  1. Show by example that the radius of convergence of a product of power series may be more than the minimum of their radii of convergence.
  2. Is this still true if both series have finite non-zero radii of convergence?

1. Some computations

   1. Find $x, y \in \R$ so that $x + iy$ is a primitive 16th root of unity. Express your answer using rationals and radicals. There are multiple correct answers to this problem, as there is more than one primitive 16th root of unity. You may without proof any of the identities and values of trigonometric functions present on the linked Wikipedia page List of trigonometic functions (as of 21:21, 15 January 2020). You will notice that this includes the values of $\sin$ and $\cos$ on angles of the form $\frac{k\pi}{n}$ with $n \in \set{2, 3, 4, 6}$ but not larger, so some computation will be necessary.
   2. Prove that if $a \in \C\setminus\set0$, the equation $z^n=a$ has exactly $n$ distinct solutions in $\C$. (You may find it useful that $\sin$ and $\cos$ are both $2\pi$-periodic.)
   3. Compute all solutions of the equation $z^6 = -\frac{27}2 + i\frac{27\sqrt3}2$. Express your answer(s) in the form $r\cis(\theta)$.
   4. Let $n \gt 1$ and let $\zeta \neq 1$ be an $n$-th root of unity. Prove that \[\sum_{k=0}^{n-1}\zeta^k = 0.\]

2. Some useful facts about holomorphic functions

   Let $\Omega \subseteq \C$ be open, and $z \in \Omega$.
   1. Suppose that $f : \Omega \to \C$ is holomorphic at $z$. Prove that $f$ is continuous at $z$.
   2. Prove the product rule: show that if $f, g : \Omega \to \C$ are holomorphic at $z$, then $fg$ is holomorphic at $z$ with \[(fg)'(z) = f'(z)g(z) + f(z)g'(z).\]

3. Some counterexamples

   Give examples of functions $f = u + iv$ defined on $\C$ so that:
   1. $f$ is holomorphic at $0$ but no other point in $\C$;
   2. $f$ satisfies the Cauchy-Riemann equations at $0$ and is continuous at $0$, but is not holomorphic at $0$;
   3. $f$ is holomorphic at $0$, the partial derivatives of $u$ with respect to $x$ and $y$ exist in a neighbourhood of $0$, but at least one of those partial derivatives fails to be continuous at $(0,0)$. (A set $S \subseteq \C$ is called a "neighbourhood of $z_0$" if $S$ is open and $z_0 \in S$.)
   (You may find the function $g : \R \to \R$ given by \[g(t) = \begin{cases}0 & \text{ if } t = 0 \\ t^2\sin\paren{\frac1t} & \text{ otherwise}\end{cases}\] to be useful.)

4. A fact that will be useful for power series

   Recall that for a sequence $(a_n)_n$ in $\R$, its limit superior is defined by \[\limsup_{n\to\infty} \, a_n := \lim_{n \to \infty} \paren{\sup\set{a_k \mid k \gt n}} \in \R\cup\set{\pm\infty}.\] The limit superior of a sequence always exists, and—if the sequence is bounded—is finite. Before proceeding to the following problem, spend some time to convince yourself that $\displaystyle\limsup_{n\to\infty} \, a_n$ is the unique $x \in \R\cup\set{\pm\infty}$ with the following two properties:
   • if $y \gt x$, then $a_n \gt y$ for only finitely many $n$; and
   • if $y \lt x$, then $a_n \gt y$ infinitely often.

   Suppose that $(c_n)_n$ is a sequence in $\C$, and define \[R = \sup\set{r \geq 0 \mid (r^n c_n)_n \text{ is bounded}}.\]

   Show that if $R \gt 0$ then \[\limsup_{n\to\infty} \abs{c_n}^{1/n} = 1/R,\] interpreting $1/\infty$ as $0$. Also show that if $R = 0$ if and only if the sequence $\paren{\abs{c_n}^{1/n}}_n$ is unbounded.

   You may be cavalier with the use of standard limits such as \[\limni A^{\frac1n} = 1 \qquad\text{ for } A \in (0, \infty),\] if you find them helpful.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• A problem related to problem 2 and a difficult-to-read blackboard

  Suppose $\Omega \subseteq \C$ is open and $z \in \Omega$. Characterize which $f, g : \Omega \to \C$ are holomorphic at $z$ and satisfy \[(fg)'(z) = f'(z)g(z) - f(z)g'(z)\]

• A proof of a lemma from class

  Suppose $\Omega \subseteq \C$ is open and $f : \Omega \to \C$. Fix $z \in \Omega$.

  Show that $f$ is holomorphic at $z$ if and only if there is $a \in \C$ and a function $E : \C \to \C$ so that: \[f(z+h) = f(z) + ah + hE(h) \qquad\text{and}\qquad \lim_{h\to0} E(h) = 0.\]

• The remaining rules of calculus

  Suppose that $\Omega, U \subseteq \C$ are open, and fix $z \in \Omega$.
  1. Prove that the derivative is linear: that is, if $f, g : \Omega \to \C$ are holomorphic at $z$ and $\lambda \in \C$, then $f+\lambda g$ is holomorphic at $z$ with \[(f+\lambda g)'(z) = f'(z) + \lambda g'(z).\]
  2. Prove the chain rule: if $f : \Omega \to U$ and $g : U \to \C$ are holomorphic at $z$ and $f(z)$ respectively, then $g\circ f$ is holomorphic at $z$ with \[(g\circ f)'(z) = g'(f(z))f'(z).\]
  3. Let $q : \C\setminus\set0 \to \C$ be given by $q(z) = \frac1z$. Show that $q$ is holomorphic on its domain, with \[q'(z) = -\frac1{z^2}.\] Deduce from this and the other rules of differentiation the quotient rule: if $f, g : \Omega \to \C$ are holomorphic at $z$ and $g(z) \neq 0$, then $f/g$ is holomorphic at $z$ with \[\paren{\frac fg}'(z) = \frac{f'(z)g(z) - f(z)g'(z)}{g(z)^2}.\]

• An extension of problem 4

  Within the context of problem 4, suppose that \[\lim_{n\to\infty}\abs{\frac{c_{n+1}}{c_n}} = L.\] Show that $L = 1/R$.