Math 185 Fall 2015 - Complex Analysis

Mondays, Wednesdays, Fridays 3pm-4pm 3 Evans
Instructor: David Dynerman dynerman@math.berkeley.edu
David's Office Hours: Mondays 4pm-5pm
(all in 943 Evans) Wednesdays 4pm-5pm
  Fridays 11am-12pm
  …or email for an appointment
Christopher's (GSI) Office Hours: Mondays 10:30am-12:30pm
(all in 1039 Evans) Tuesdays 4:30pm-6:30pm
  Wednesdays 10:30am-12:30pm
  Thursdays 4:30pm-6:30pm
Text: StSh Complex Analysis, Stein & Shakarchi
  S Complex Function Theory, Sarason (optional)

Table of Contents

Final Exam Study Guide

Here is some review information for the final exam. Here is a suggested way to study:

  1. Go through the "Concepts" section below with your study group. For each concept, review your notes and the book.
  2. Go through all your homework assignments. Review your solutions. Make a list of problems you didn't completely understand.
  3. Work on the problems you didn't completely understand. Email questions, ask in the review sessions or Piazza.
  4. Work through the suggested extra problems.

Concepts

  1. Basics
    1. Basics of calculus in \( \mathbb{C} \): Taking limits, differentiating functions, computing integrals (parametrizing paths)
    2. Basic properties of holomorphic functions (Cauchy-Riemann equations, power series)
    3. What is the difference between a differentiable function \( \mathbb{R}^2 \to \mathbb{R}^2 \) and a holomorphic function \( \mathbb{C} \to \mathbb{C} \)?
  2. Holomorphic functions
    1. Goursat's theorem and Cauchy's theorem for a disc
    2. What's the deal with antiderivatives of holomorphic functions? Do they always exist? How does Cauchy's theorem for the disc let us construct antiderivatives in discs?
    3. Using contour integration and Cauchy's theorem to get formulas for real integrals
    4. Cauchy integral formulas, Liouville's theorem
    5. What is the behavior of the zeros of a holomorphic function?
    6. Analytic continuation of holomorphic functions
    7. When do sequences of holomorphic functions converge to a holomorphic function? What happens with the derivatives?
    8. Schwarz reflection principle
  3. Meromorphic functions
    1. What is the definition of an isolated singularity? What does it mean for a holomorphic function to have a singularity at \( z_0 \)?
    2. Types of isolated singularities
    3. What is the precise meaning for a function to have a zero of a certain order? A pole of a certain order?
    4. Residue formula
    5. Using contour integration and the Residue Theorem to obtain formulas for real integrals ("Calculus of residues")
    6. Laurent Series for holomorphic functions with singularities. What do they look like for essential singularities? Poles? Removable singularities?
    7. Argument principle and applications: Open mapping theorem, maximum modulus principle
  4. Cauchy's theorem in general
    1. Homotopies and simple connectedness. Examples of simply connected and not-simply-connected domains in \( \mathbb{C} \).
    2. If two paths with common endpoints are homotopic, then the integral of a holomorphic function along both paths is the same. How can we use this to establish Cauchy's theorem for general simply connected domains?
    3. How can we use Cauchy's theorem for simply connected domains to obtain antiderivatives on such domains?
  5. Complex Logarithm
    1. What's the deal with the complex logarithm? Does it always exist? Is it always holomorphic? Entire?
    2. When does a complex log exist?
    3. When it does exist on some fixed domain, is it always unique? Is it unique if 1 is in our domain and we choose the branch with \(\Log 1 = 0\)? Why?
  6. Conformal mappings
    1. What is a conformal map? What does it mean for two open sets to be biholomorphic?
    2. Examples of conformal maps and conformal equivalences in \( \mathbb{C} \)
    3. Schwarz lemma
    4. What is the automorphism group of the disc? What is the automorphism group of the upper half-plane?
    5. Riemann mapping theorem
  7. Fourier transforms
    1. Definition of Fourier transforms
    2. Computations of a few simple Fourier transforms using contour integration
  8. Gamma and Zeta functions
    1. Basic properties - when/where are they holomorphic/meromorphic? What are their formulas?

Last homework: Extra Review Problems

Here are extra problems that you should study for the final exam. These are in addition to your homework problems, which you should look at first.

These extra problems are your last homework for the semester. Please bring your solutions to the final - as long as you've made a reasonable attempt on a good chunk of them, you'll receive credit.

The problems are marked by priority i) very relevant, start here; ii) (⌐■_■) it'd be good to know these; iii) (ー。ー) good problems, but focus on category i) and ii) first.

  1. Basics
    1. StSh (⌐■_■) 1.4.9, 1.4.11, (⌐■_■) 1.4.20, 1.4.24
  2. Holomorphic functions
    1. StSh 2.6.2, 2.6.3, 2.6.4, (⌐■_■) 2.6.6, (⌐■_■) 2.6.7, (⌐■_■) 2.6.13, 2.6.14, (ー。ー) 2.6.15
  3. Meromorphic functions/Cauchy's theorem in general/Complex logarithm
    1. StSh 3.8.4, 3.8.6, 3.8.7, 3.8.8, 3.8.12, (⌐■_■) 3.8.12, (ー。ー) 3.8.13, (ー。ー) 3.8.17
  4. Conformal mappings
    1. StSh 8.5.5, 8.5.9, (⌐■_■) 8.5.10, (ー。ー) 8.5.14
  5. Fourier transforms
    1. StSh 4.4.6, Chapter 2 Example 1, Chapter 3 Example 3
  6. Gamma & Zeta Functions
    1. StSh (ー。ー) 6.3.1, (ー。ー) 7.3.3

Homework

HW 7, Due November 20

  1. (StSh 4.4.3) Show, by contour integration, that if \( a > 0 \) and \( \xi \in \mathbb{R} \), then \begin{equation*} \frac{1}{\pi} \int_{-\infty}^\infty \frac{a}{a^2 + x^2} e^{-2\pi i x \xi} \, dx = e^{-2\pi a | \xi |}, \end{equation*}

    and check that

    \begin{equation*} \int_{-\infty}^\infty e^{-2\pi a | \xi | }e^{2\pi i \xi x}\, d \xi = \frac{1}{\pi} \frac{a}{a^2 + x^2}. \end{equation*}

    In other words, in this problem you show that the Fourier transform of \( \frac{a}{a^2 + x^2} \) is \( \pi e^{-2\pi a |\xi|} \) and check that the inverse Fourier transform of \( \pi e^{-2\pi a | \xi |} \) returns to \( \frac{a}{a^2 + x^2} \).

  2. (StSh 4.4.4) Suppose \( Q \) is a polynomial of degree \( \geq 2 \) with distinct roots, none lying on the real axis. Calculate \begin{equation*} \int_{-\infty}^\infty \frac{e^{-2\pi i x \xi}}{Q(x)} \, dx, \qquad \xi \in \mathbb{R} \end{equation*}

    in terms of the roots of \( Q \). What happens when several roots coincide?

    [Hint: Consider separately the cases \( \xi < 0, \xi = 0, \) and \( \xi > 0 \). Use residues.]

  3. (StSh 8.5.3) Suppose \( U \) and \( V \) are conformally equivalent. Prove that if \( U \) is simply connected, then so is \( V \). Note that this conclusion remains valid if we merely assume that there exists a continuous bijection between \( U \) and \( V \).
  4. (StSh 8.5.8) Recall that a function \( u(x,y) \) on \( \mathbb{R}^2 \) is harmonic if \( \nabla u = 0 \), where \( \nabla \) denotes the Laplacian, \begin{equation*} \nabla = \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}. \end{equation*}

    Find a harmonic function \( u \) in the open first quadrant that extends continuously up to the boundary except at the points \( 0 \) and \( 1 \), and that takes on the following boundary values: \( u(x, y) = 1 \) on the half-lines \( \{ y = 0, x > 1 \} \) and \( \{ x = 0, y > 0 \} \), and \( u(x,y) = 0 \) on the segment \( \{ 0 < x < 1, y = 0 \} \).

    [Hint: Find conformal maps \( F_1, F_2, \ldots, F_5 \) indicated in Figure 1. Note that \( \frac{1}{\pi} \arg(z) \) is harmonic on the upper half-plane, equals \( 0 \) on the positive real axis, and \( 1 \) on the negative real axis.]

    hw7_conformal.png

    Figure 1: Sequence of conformal maps for Problem 4.

  5. (StSh 8.5.15) Here are two properties enjoyed by automorphisms of the upper half-plane.
    1. Suppose \( \Phi \) is an automorphism of \( \mathbb{H} \) that fixes three distinct points on the real axis. Prove that \( \Phi \) is the identity.
    2. Suppose \( (x_1, x_2, x_3) \) and \( (y_1, y_2, y_3) \) are two pairs of three distinct points on the real axis with \begin{equation*} x_1 < x_2 < x_3 \qquad \text{and} \qquad y_1 < y_2 < y_3. \end{equation*}

      Prove that there exists a unique automorphism \( \Phi \b) of \( \mathbb{H} \) so that \( \Phi(x_j) = y_j, j = 1, 2, 3\).

      [Note: The same conslusion holds if \( y_3 < y_1 < y_2 \) or \( y_2 < y_3 < y_1 \).]

HW 6, Due October 30

  1. Find and classify (i.e., removable, pole or essential) all the isolated singularities \( z_i \) for the following functions. If a singularity \( z_i \) is removable, compute the value required to make the function holomorphic in a neighborhood of \( z_i \). If a singularity \( z_i \) is a pole, compute the residue of \( f \) at \( z_i \). If a singularity \( z_i \) is essential, run out of the room flailing your arms wildly. Justify your answers.
    1. \begin{equation*} f(z) = \frac{1 + \cos(z)}{(z - \pi)^2} \end{equation*}
    2. \begin{equation*} f(z) = \frac{\sin(z)}{z^3} \end{equation*}
    3. \begin{equation*} f(z) = \sin(1/z) \end{equation*}
  2. Using contour integration,
    1. (StSh 3.8.9) show \begin{equation*} \int_0^1 \log(\sin \pi x) \, dx = - \log 2; \end{equation*}

      [Hint: Use the contour in Figure 2.]

      hw6_contour.png

      Figure 2: Contour for the Problem 2.1.

    2. (StSh 3.8.11) assuming \( |a| < 1 \), show \begin{equation*} \int_0^{2\pi} \log | 1 - ae^{i\theta} | \, d\theta = 0; \end{equation*}
    3. evaluate the definite integral \begin{equation*} \int_0^{2\pi} \frac{d\theta}{3 - 2\cos(\theta)}. \end{equation*}

      [Hint: Change coordinates via \( z = e^{i\theta} \) and apply an identity to simplify the integrand.]

  3. (StSh 3.8.18) Give another proof of the Cauchy integral formula: \begin{equation*} f(z) = \frac{1}{2\pi i} \int_C \frac{f(\xi)}{\xi - z} \, d\xi \end{equation*}

    using homotopy of curves.

    [Hint: Deform the circle \( C \) to a small circle centered at \( z \), and note that the quotient \( (f(\xi) - f(z))/(\xi - z) \) is bounded.]

  4. (StSh 3.9.3) If \( f(z) \) is holomorphic in the deleted neighborhood \( \{ 0 < | z - z_0 | < r \} \) and has a pole of order \( k \) at \( z_0 \), then we can write \begin{equation*} f(z) = \frac{a_{-k}}{(z - z_0)^k} + \cdots + \frac{a_{-1}}{(z - z_0)} + g(z), \end{equation*}

    where \( g \) is holomorphic in the disc \( \{ | z - z_0 | < r \} \). There is a generalization of this expansion that holds even if \( z_0 \) is an essential singularity. This is a special case of the Laurent series expansion, which is valid in an even more general setting.

    Let \( f \) be holomorphic in a region containing the annulus \( \{ z : r_1 \leq | z - z_0 | \leq r_2 \} \), where \( 0 < r_1 < r_2. \) Then, show that

    \begin{equation*} f(z) = \sum_{n = -\infty}^\infty a_n(z - z_0)^n \end{equation*}

    where the series converges absolutely on the interior of the annulus.

    [Hint: To prove this, it suffices to write

    \begin{equation*} f(z) = \frac{1}{2\pi i} \int_{C_{r_2}} \frac{f(\xi)}{\xi - z} \, d\xi - \frac{1}{2\pi i} \int_{C_{r_1}} \frac{f(\xi)}{\xi - z} \, d\xi \end{equation*}

    when \( r_1 < | z - z_0 | < r_2 \), and argue as in the proof of Theorem 4.4, Chapter 2 StSh. Here \( C_{r_1} \) and \( C_{r_2} \) are the circles bounding the annulus.]

  5. (StSh 3.8.21)

    Definition. An open set \( \Omega \subset \mathbb{C} \) is star-shaped if there exists a point \( z_0 \in \Omega \) such that for any \( z \in \Omega \) the straight line segement between \( z \) and \( z_0 \) is contained in \( \Omega \).

    Prove that any star-shaped domain is simply connected.

    Note: This gives another proof that the slit plane \( \mathbb{C} - \{ (-\infty, 0] \} \) is simply connected.

HW 5, Due October 19

  1. (StSh 2.7.3) Morera's theorem states that if \( f \) is continuous in \( \mathbb{C} \), and \( \int_T f(z) \, dz = 0 \) for all triangles \( T \), then \( f \) is holomorphic in \( \mathbb{C} \).

    In this problem you will prove a similar theorem, replacing "all triangles" with "all circles":

    Theorem: Suppose \( f \) is continuous in \( \mathbb{C} \), and \( \int_C f(z) \, dz = 0 \) for all circles \( C \). Then \( f \) is holomorphic in \( \mathbb{C} \).

    1. Prove the theorem in the special case where \( f \) is twice-differentiable as a function of the real variables \( x \) and \( y \).

      Hint: Write \( f(z) = f(z_0) + a(z-z_0) + b(\overline{z-z_0}) + O(|z-z_0|^2) \) for \( z \) near \( z_0 \). By integrating this expansion over small circles centered at \( z_0 \), conclude \( \partial f/\partial \overline{z} = b = 0 \) at \( z_0 \).

    2. Prove the theorem in the general case, where \( f \) is only assumed to be continuous.

      Hint: Approximate \( f \) by smooth functions, and then apply your result from part 1 above. More precisely, let \( \phi(x,y) \) be a smooth function on \( \mathbb{R}^2 \) such that \( 0 \leq \phi(x,y) \leq 1 \) and \( \int_{\mathbb{R}^2} \phi(x,y) \, dx dy = 1 \).

      Then, for \( \epsilon > 0 \), let \( \phi_\epsilon(x,y) = \epsilon^{-2}\phi(\epsilon^{-1}x,\epsilon^{-1}y) \), and

      \begin{equation*} f_\epsilon(x_0,y_0) = \int_{\mathbb{R}^2} f(x_0-x,y_0-y)\phi_\epsilon(x,y) \, dx\,dy. \end{equation*}

      Show that \( f_\epsilon \to f \) uniformly on any compact subset of \( \mathbb{C} \) and that each \( f_\epsilon \) is smooth and satisfies the hypothesis of the theorem. Applying your part 1 above and Theorem 5.2 in Chapter 2 of StSh, conclude that \( f \) is holomorphic.

  2. (StSh 3.8.3) Show that \begin{equation*} \int_{-\infty}^\infty \frac{\cos x}{x^2 + a^2} \, dx = \pi \frac{e^{-a}}{a}, \qquad \text{ for $a > 0$}. \end{equation*}
  3. (S VII.12.2) Determine the residues of the following functions at each of their isolated singularities in \( \mathbb{C} \):
    1. \begin{equation*} \frac{z^p}{1 - z^q}, \qquad p, q \text{ positive integers}, \end{equation*}
    2. \begin{equation*} \frac{z^5}{(z^2 - 1)^2}, \end{equation*}
    3. \begin{equation*} \frac{\cos z}{1 + z + z^2}, \end{equation*}
    4. \begin{equation*} \frac{1}{\sin z}. \end{equation*}
  4. (StSh 3.8.14) Prove that all entire functions that are also injective are linear, e.g., they take the form \( f(z) = az + b \) with \( a, b \in \mathbb{C} \) and \(a \neq 0 \).

    Hint: Apply the Casorati-Weierstrass theorem to \( f(1/z) \).

  5. (StSh 3.8.15) Use the Cauchy inequalities or the maximum modulus principle to solve the following problems:
    1. Prove that if \( f \) is an entire function that satisfies \begin{equation*} \sup_{|z| = R} |f(z)| \leq AR^k + B \end{equation*}

      for all \( R > 0 \), and for some integer \( k \geq 0 \) and some constants \( A, B > 0 \), then \( f \) is a polynomial of degree \( \leq k \).

    2. Let \( w_1, \ldots, w_n \) be points on the complex unit circle. Prove that there exists a point \( z \) on the unit circle such that the product of the distances from \( z \) to the points \( w_j, 1 \leq j \leq n \), is at least 1. Conclude that there exists a point \( w \) on the unit circle such that the product of the distances from \( w \) to the \( w_j, 1 \leq j \leq n \) is exactly 1.

HW 4, Due October 5

  1. (StSh 2.6.1) Prove that \begin{equation*} \int_0^\infty \sin(x^2) \, dx = \int_0^\infty \cos(x^2) \, dx = \frac{\sqrt{2\pi}}{4}. \end{equation*}

    These are the Fresnel integrals. Here, \( \int_0^\infty \) is interpreted as \( \lim_{R \to \infty} \int_0^R \).

    [Hint: Integrate the function \( e^{-z^2} \) over the contour below. Recall that \( \int_{\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi} \).]

    hw4_fresnel_contour.png

    Figure 3: Contour for the Fresnel integrals

  2. (StSh 2.6.2) Show that \begin{equation*} \int_0^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}. \end{equation*}

    [Hint: The integral equals \( \frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{e^{ix} - 1}{x} \, dx \). Use the indented semicircle.]

  3. (S VII.4.2) Let \( a \) be a positive number. Evaluate the integrals \begin{equation*} \int_0^\infty \frac{1}{t^4 + a^4} \, dt, \qquad \qquad \int_0^\infty \frac{t^2}{t^4 + a^4} \, dt \end{equation*}

    by integrating the function \( (z^2 + a^2)^{-1} \) around the triangle with vertices \( 0, R, Re^{\frac{\pi i}{4}} (R > 0) \) and taking the limit as \( R \to \infty \).

  4. (StSh 2.6.7) Suppose \( f: \mathbb{D} \to \mathbb{C} \) is holomorphic. Show that the diameter \( d = \sup_{z, w \in \mathbb{D}} | f(z) - f(w) | \) of the image satisfies \begin{equation*} 2|f'(0)| \leq d. \end{equation*}

    [Hint: \( 2f'(0) = \frac{1}{2\pi i} \int_{|\xi| = r} \frac{f(\xi) - f(-\xi)}{\xi^2} \, d\xi \) whenever \( 0 < r < 1\).]

  5. (StSh 2.6.9) Let \( \Omega \) be a bounded open subset of \( \mathbb{C} \), and \( \phi: \Omega \to \Omega \) a holomorphic function. Prove that if there exists a point \( z_0 \in \Omega \) such that \begin{equation*} \phi(z_0) = z_0 \qquad \text{and} \qquad \phi'(z_0) = 1 \end{equation*}

    then \( \phi \) is linear.

    [Hint: Why can one assume that \( z_0 = 0 \)? Write \( \phi(z) = z + a_nz^n + O(z^{n+1}) \) near 0, and prove that if \( \phi_k = \phi \circ \phi \circ \ldots \circ \phi \) (where \( \phi \) appears \( k \) times), then \( \phi_k(z) = z + ka_nz^n + O(z^{n+1}) \). Apply the Cauchy inequalities and let \( k \to \infty \) to conclude the proof. Here we use the standard \( O \) ("big-Oh") notation, where \( f(z) = O(g(z)) \) as \( z \to 0 \) means that \( | f(z) | \leq C | g(z) | \) for some constant \( C \) as \( | z | \to 0 \)].

HW 3, Due September 18

  1. Suppose \( a_n \) is a sequence of real numbers with \( \lim \sup a_n = L \). If \( \epsilon > 0 \), prove that the sequence eventually stays below \( L + \epsilon \), that is, show that there is a natural number \( N \) such that \begin{equation*} a_n < L + \epsilon \end{equation*}

    for all \( n > N \).

  2. (S IV.9.3) In what sense is it true that \( \log a_1a_2 = \log a_1 + \log a_2 \) for \( a_1, a_2 \in \mathbb{C} \)?
  3. (StSh 1.23) Consider the function \( f \) defined on \( \mathbb{R} \) by \begin{equation*} f(x) = \begin{cases} 0 & \text{if } x \leq 0, \\ e^{-1/x^2} & \text{if } x > 0. \end{cases} \end{equation*}

    Prove that \( f \) is infinitely differentiable on \( \mathbb{R} \), and that \( f^{(n)}(0) = 0 \) for all \( n \geq 1 \). Conclude that \( f \) does not have a power series representing it near the origin.

  4. (S VI.7.1) Suppose that \( \gamma: [0, 2\pi] \to \mathbb{C} \) is the curve given by \begin{equation*} \gamma(\theta) = e^{i\theta}. \end{equation*}

    First, show that

    \begin{equation*} \int_\gamma \frac{1}{z} \, dz = \int_0^{2\pi} \frac{1}{\gamma(\theta)} \gamma'(\theta) \, d\theta = 2\pi i. \end{equation*}

    Recall that a branch of \( \arg \) on an open set \(\Omega\) is a continuous function \( \alpha \) such that \( \alpha(z) \) is an argument of \( z \) for all \( z \in \Omega \). Use your calculation above to deduce that there is no branch of \( \arg \) in \( \Omega = \{ z \in \mathbb{C} \, \mid \, 0 < | z | < 1 \} \).

  5. (StSh 1.26) Suppose \( f \) is continuous in a region \( \Omega \). Prove that any two primitives of \( f \) (if they exist) differ by a constant.

HW 2, Due September 11-September 18

  1. (S II.8.2) Let the function \( f \) be holomorphic in the open set \( \Omega \). Prove that the function \( g(z) = \overline{f(\overline{z})} \) is holomorphic in the set \( \overline{\Omega} = \{ \overline{z} \, \mid \, z \in G \} \).
  2. (StSh 1.15) Abel's Theorem. Suppose \( \sum_{n=1}^\infty a_n \) converges. Prove that \begin{equation*} \lim_{r \to 1, r < 1} \sum_{n=1}^\infty r^n a_n = \sum_{n=1}^\infty a_n. \end{equation*}

    [Hint: Use summation by parts - see problem StSh 1.14.] In other words, if a series converges, then it is Abel summable with the same limit. Abel summation is an alternative way to define the sum of an infinite series. There exist divergent series which are Abel summable, so Abel summaion provides a further tool for studying such series. For more information on different ways of summing infinite series, see Stein & Shakarchi, Fourier Analysis: An Introduction, Chapter 2.

  3. (StSh 1.16) Determine the radius of convergence of the series \( \sum_{n=1}^\infty a_n z^n \) when:
    1. \( a_n = (\log n)^2 \)
    2. \( a_n = n! \)
    3. \( a_n = \frac{n^2}{4^n + 3n} \)
    4. \( a_n = (n!)^3 / (3n)! \)

      [Hint: Use Stirling's formula, which says \( n! \sim cn^{n+\frac{1}{2}}e^{-n} \) for some \( c > 0 \). The notation \( A(n) \sim B(n) \) (read "\( B(n) \) is an asymptotic formula for \( A(n) \)") means that limn→ ∞ A(n)/B(n) = 1.]

    5. Find the radius of convergence of the hypergeometric series \begin{equation*} F(\alpha, \beta, \gamma; z) = 1 + \sum_{n=1}^\infty \frac{\alpha(\alpha + 1)\cdots (\alpha + n - 1)\beta(\beta+1)\cdots (\beta + n-1)}{n!\gamma(\gamma + 1) \cdots (\gamma + n - 1)} z^n. \end{equation*}

      Here \( \alpha, \beta, \gamma \in \mathbb{C} \) and \( \gamma \in \mathbb{R} \) with \( \gamma \neq 0, -1, -2, \ldots \)

    6. Find the radius of convergence of the Bessel function of order \( r \): \begin{equation*} J_r(z) = \left ( \frac{z}{2} \right )^r \sum_{n=0}^\infty \frac{(-1)^n}{n!(n+r)!}\left( \frac{z}{2} \right )^{2n}, \end{equation*}

      where \( r \) is a positive integer.

      Note: The Bessel functions appear when studying several differential equations that are important in physics. They also relate the Fourier coefficients of a function and its Fourier transform.

  4. (StSh 1.19) Prove the following:
    1. The power series \( \sum nz^n \) does not converge on any point of the unit circle.
    2. The power series \( \sum z^n/n^2 \) converges at every point of the unit circle.
    3. The power series \( \sum z^n/n \) converges at every point of the unit circle except \( z = 1 \). [Hint: Sum by parts].

    Note: Recall that the unit circle is the set of complex numbers at distance \( 1 \) from the origin, \( \{ z \in \mathbb{C} \, \mid \, |z| = 1 \} \).

  5. (StSh 1.22) Let \( \mathbb{N} = \{ 1, 2, 3, \ldots \} \) denote the set of positive integers. A subset \( S \subset \mathbb{N} \) is said to in arithmetic progression if \begin{equation*} S = \{ a, a + d, a + 2d, a + 3d, \ldots \} \end{equation*}

    where \( a, d \in \mathbb{N} \). Here \( d \) is called the step of \( S \).

    Show that \( \mathbb{N} \) cannot be partitioned into a finite number of subsets that are in arithmetic progression with distinct steps (except for the trivial case \( a = d = 1 \)).

    [Hint: Write \( \sum_{n \in \mathbb{N}} z^n \) as a sum of terms of the type \( \frac{z^a}{1 - z^d} \).]

HW 1, Due September 4

  1. (S I.9.1) Prove that \( \arg \overline{z} = \arg z^{-1} = -\arg z \) for any non-zero complex number \( z \).
  2. (StSh 1.7) The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters.
    1. Let \( z, w \) be two complex numbers such that \( \overline{z}w \neq 1 \). Prove that \begin{equation*} \left | \frac{w - z}{1 - \overline{w}z} \right | < 1 \qquad \text{if } |z| < 1 \text{ and } |w| < 1, \end{equation*}

      and also that

      \begin{equation*} \left | \frac{w - z}{1 - \overline{w}z} \right | = 1 \qquad \text{if } |z| = 1 \text{ or } |w| = 1. \end{equation*}

      [Hint: Why can one assume that z is real? It then suffices to prove that

      \begin{equation*} (r-w)(r - \overline{w}) \leq (1 - rw)(1 - r\overline{w}) \end{equation*}

      with equality for appropriate \( r \) and \( |w| \).]

    2. Prove that for a fixed \( w \) in the unit disc \( \mathbb{D} \), the mapping \begin{equation*} F : z \mapsto \frac{w-z}{1 - \overline{w}z} \end{equation*}

      satisfies the following conditions:

      1. \( F \) maps the unit disc to itself (that is, \( F: \mathbb{D} \to \mathbb{D} \)), and is holomorphic.
      2. \( F \) interchanges \( 0 \) and \( w \), namely \( F(0) = w \) and \( F(w) = 0 \).
      3. \( |F(z)| = 1 \) if \( |z| = 1 \).
      4. \( F: \mathbb{D} \to \mathbb{D} \) is bijective. [Hint: Calculate \( F \circ F \)]
  3. (StSh 1.8) Suppose \( U \) and \( V \) are open sets in the complex plane. Prove that if \( f: U \to V \) and \( g: V \to \mathbb{C} \) are two functions that are differentiable (in the real sense, that is, as functions of the two real variables \( x \) and \( y \)), and \( h = g \circ f \), then \begin{equation*} \frac{\partial h}{\partial z} = \frac{\partial g}{\partial z} \frac{\partial f}{\partial z} + \frac{\partial g}{\partial \overline{z}} \frac{\partial \overline{f}}{\partial z} \end{equation*}

    and

    \begin{equation*} \frac{\partial h}{\partial \overline{z}} = \frac{\partial g}{\partial z} \frac{\partial f}{\partial \overline{z}} + \frac{\partial g}{\partial \overline{z}} \frac{\partial \overline{f}}{\partial \overline{z}}. \end{equation*}

    This is the complex version of the chain rule.

  4. (StSh 1.12) Consider the function defined by \begin{equation*} f(x + iy) = \sqrt{|x||y|}, \qquad \text{whenever } x, y \in \mathbb{R}. \end{equation*}

    Show that \( f \) satisfies the Cauchy-Riemann equations at the origin, yet \( f \) is not holomorphic at \( 0 \).

  5. (S II.8.1) Let the function \( f \) be holomorphic in the open disk \( \mathbb{D} \). Prove that each of the following conditions forces \( f \) to be constant:
    1. \( f' = 0 \) throughout \( \mathbb{D} \);
    2. \( f \) is real-valued in \( \mathbb{D} \);
    3. \( |f| \) is constant in \( \mathbb{D} \);
    4. \( \arg f \) is constant in \( \mathbb{D} \);

Schedule

Date   Read before class
Aug 26 basic definitions & arithmetic; the complex plane; absolute value and conjugation; the triangle inequality  
Aug 28 polar form; geometry of multiplication; De Moivre's formula and roots of unity; limit points, interior points; open & closed sets; StSh 1.1.1-1.1.3, S I.1-I.10
Aug 31 limits in \( \mathbb{C} \); continuous functions; holomorphic functions; StSh 1.2.1-1.2.2, S II.1-II.8
Sep 2 derivatives as linear approximations; holomorphic functions as maps \( \mathbb{R}^2 \to \mathbb{R}^2 \); Cauchy-Riemann equations; examples  
Sep 4 HW 1 Due; differentiability in \( \mathbb{R}^2 \)  
Sep 7 No Class, Labor Day  
Sep 9 sequences & series; power series; radius of convergence StSh 1.2.2, 1.2.3, S V.1-V.3
Sep 11 HW 2 Due; computing the radius of convergence; holomorphicity of power series S V.4-V.11
Sep 14 differentiating power series; ratio test; power series examples; trigonometric series; logarithms StSh 1.2.3,1.3, S V.12-V.15, IV.8-IV.12
Sep 16 integration in \( \mathbb{C} \), fundamental theorem of calculus; triangle inequality; integration along curves S VI.1-VI.5
Sep 18 HW 3 Due; Midterm 1 review All of the above!
Sep 21 Midterm 1  

How to learn a lot

The simple advice is to stay engaged with the course:

  • read the indicated sections before coming to lecture
  • start homework assignments early and work on problems with others
  • review your notes after class
  • when you don't understand something, please email, ask in class, or come to office hours
  • when you get stuck on a homework problem, please email, ask in class, or come to office hours
  • email questions, ask questions in class, come to office hours

Course Policies

Grading

Your course grade will consist of two midterms and a final exam, homework sets, and unannounced 5 minute quizzes on definitions we've covered in class. Your final grade will be weighted as follows:

Midterm 1 Sept. 21 (in class) 20%
Midterm 2 Nov. 2 (in class) 20%
Final Exam Dec. 15 (7pm-10pm) 30%
Homework     20%
Quizzes   (lowest score dropped) 10%

Homework solutions

I encourage you to work with others to solve the homework problems, but you must submit your own individual write-up. Indicate on your solutions who you collaborated with, and cite any sources used (other books, internet Q&A sites, Google, etc).

You should hand in a neat, final draft of your solution. Proofs must explain the argument in complete English sentences.

Missed exams, quizzes and late homework

If you have a conflict with one of the exam times due to religious or extracurricular reasons, please see me during the first two weeks of the semester to make alternate arrangements.

Homework is due in class on the day indicated. In general late work will not be accepted because it's a burden on the grader, and because we plan to distribute solutions. Similarly quizzes are meant to be a quick check that you're keeping up with material, so make up quizzes don't make a lot of sense. The lowest quiz grade will be dropped, and there will be enough of them that missing a handful will not substantially change your grade.

If this is causing you a lot of grief, please talk to me.