Week 2 Worksheet

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Functions

1. What are the real and imaginary parts of $f(z) = \Re(z)$ and $f(z) = |z|$?
2. Let $f(z) = z^2$ on $\C$.
   1. Determine the real and imaginary parts of $f$.
   2. Calculate the gradients of $u$ and $v$. (Recall that \[ (\nabla h)(x,y) = \Big\langle (\partial_1 h)(x,y), (\partial_2 h)(x,y) \Big\rangle \] is the gradient of $h : \R^2 \to \R$.)
   3. Verify that $\nabla u$ and $\nabla v$ are perpendicular.
3. Let $f(z) = 1/z$ on $\C \setminus \{0\}$.
   1. What are the real and imaginary parts of $f$?
   2. What are the level curves of the real part?
4. Prove that $f(z) = 1/z$ is continuous on $\C \setminus \{0\}$.
5. For each of the following, determine the limit or explain why it does not exist.
   1. $\lim\limits_{z \to 0} \dfrac{|z|}{z}$
   2. $\lim\limits_{z \to 0} \dfrac{|z|^2}{z}$
   3. $\lim\limits_{z \to 0} \Arg(z)$
6. Prove that the following functions are continuous.
   1. $f(z) = \Re(z)$ on $\C$
   2. $f(z) = |z|$ on $\C$
   3. $f(z) = \Arg(z)$ on $\C \setminus (-\infty,0]$

Holomorphic Functions

7. Use the definition of the derivative to differentiate each of the following functions.
   1. $f(z) = z^2 + z$
   2. $f(z) = \dfrac{1}{z}$
8. 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

9. 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 $\partial_2 h$.
10. Verify that the real and imaginary parts of $f(z) = 1/z$ together satisfy the Cauchy-Riemann equations.
11. Use the Cauchy-Riemann equations to determine whether there are any $z \in \C$ where $f(z) = |z|$ is differentiable.
12. 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 $f : \C \to \C$.
13. 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 $f : \C \setminus \{0\} \to \C$.

14. 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 harmonic.)

15. 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.

16. 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.)

17. 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 $f : \C \to \C$.
18. Show that if $f : \C \to \C$ is holomorphic and $f$ has a constant real part then $f$ is constant.
19. 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 $a \in \C$.
20. 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.