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Week 3 Worksheet

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Power Series

  1. Determine the radius of convergence of each of these power series.
    1. $\displaystyle \sum_{n=0}^\infty \frac{2^n}{n} z^n$
    2. $\displaystyle \sum_{n=0}^\infty n! z^n$
    3. $\displaystyle \sum_{n=0}^\infty n^p z^n$ for a fixed $p \in \N$.
  2. The zeroth Bessel function $J_0(z)$ is defined by \[ J_0(z) = \sum_{n=0}^\infty (-1)^n \frac{1}{(n!)^2} \frac{z^n}{2n} \] but what is its radius of convergence?
  3. The power series \[ \sum_{n=0}^\infty a_n z^n \] and \[ \sum_{n=0}^\infty b_n z^n \] have radii of convergence $R > 0$ and $S > 0$ respectively. What is the radius of convergence of the series $\displaystyle \sum_{n=0}^\infty (a_n + b_n) z^n$?

Holomorphic Functions

  1. Use the definition of the derivative to differentiate each of the following functions.
    1. $f(z) = z^2 + z$
    2. $f(z) = \dfrac{1}{z}$
  2. Define $f : \C \to \C$ by $f(z) = |z|^2$.
    1. Verify that $f$ is differentiable at the origin.
    2. Is $f$ holomorphic on any domain?

Cauchy-Riemann Equations

  1. If $h : \R^2 \to \R$ is defined by $h(x,y) = 2xy$ use the definitions of the partial derivatives to calculate $\partial_1 h$ and $\partial_2 h$.
  2. Verify that the real and imaginary parts of $f(z) = 1/z$ together satisfy the Cauchy-Riemann equations.
  3. Use the Cauchy-Riemann equations to determine whether there are any $z \in \C$ where $f(z) = |z|$ is differentiable.
  4. Verify that the functions \[ u(x,y) = x^3-3xy^2 \qquad v(x,y) = 3x^2y-y^3 \] satisfy the Cauchy-Riemann equations for all $(x,y)$. Show that $u, v$ are the real and imaginary parts of a holomorphic function $f : \C \to \C$.
  5. Verify that the functions \[ u(x,y) = \frac{x^4-6x^2y^2+y^4}{(x^2+y^2)^4} \qquad v(x,y)= \frac{4xy^3-4x^3y}{(x^2+y^2)^4} \] satisfy the Cauchy-Riemann equations for all $(x,y) \ne (0,0)$. Show that $u, v$ are the real and imaginary parts of a holomorphic function $f : \C \setminus \{0\} \to \C$.
  6. Suppose that $f(z) = u(x,y)+iv(x,y)$ is holomorphic. Use the Cauchy-Riemann equations to show that both $u$ and $v$ satisfy Laplace's equation. That is, verify that \[ \partial_1 (\partial_1 u) + \partial_2 (\partial_2 u) = 0 \qquad \partial_1 (\partial_1 v) + \partial_2 (\partial_2 v) = 0 \] both hold. (The Laplacian of $g : \R^2 \to \R$ is $\triangle g = \partial_1 (\partial_1 g) + \partial_2 (\partial_2 g)$. Functions satisfying the Laplace equation $\triangle g = 0$ are called \define{harmonic}.)
  7. Let $f(z)= z^3$. Determine real-valued functions $u,v$ so that $f(x+iy) = u(x,y) + iv(x,y)$. Verify that both $u$ and $v$ satisfy the Laplace equation.
  8. Suppose $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic on $\C$. Suppose we know that $u(x,y) = x^5 - 10x^3y^2 + 5xy^4$. Use the Cauchy-Riemann equations to find all the possible forms of $v(x,y)$. (The Cauchy Riemann equations have the following remarkable implication: suppose $f(z) = u(x,y)+iv(x,y)$ is holomorphic and we know a formula for $u$. Then we can recover $v$ up to a constant; similarly, if we know $v$ then we can recover $u$ up to a constant. Hence for holomorphic functions, the real part of a function determines the imaginary part up to constants, and vice versa.)
  9. Suppose that \[ u(x,y) = x^3 -kxy^2 + 12xy - 12x \] for some constant $k \in \C$. Find all values of $k$ for which $u$ is the real part of a holomorphic function $f : \C \to \C$.
  10. Show that if $f : \C \to \C$ is holomorphic and $f$ has a constant real part then $f$ is constant.
  11. Show that the only holomorphic function $f : \C \to \C$ of the form $f(x+iy) = u(x) + iv(y)$ is given by $f(z) = \lambda z+a$ for some $\lambda \in \R$ and $a \in \C$.
  12. Suppose that $f(x+iy) = u(x,y) + iv(x,y)$. is a holomorphic function and that \[ 2 u(x,y) + v(x,y)= 5 \] for all $x + iy \in \C$. Prove that $f$ is constant.

A Cauchy-Riemann Converse

  1. Let $f(z) = \sqrt{|xy|}$ where $z=x+iy$.
    1. Show from the definition that $f$ is not differentiable at the origin.
    2. Show however that the Cauchy-Riemann equations are satisfied at the origin. Why does this not contradict our converse to the Cauchy-Riemann equations?