Week 2 Worksheet

Home | Assessment | Notes | Worksheets | Blackboard

Functions

1. What are the real and imaginary parts of $f(z)=\mathsf{Re}(z)$ and $f(z)=|z|$?
2. Let $f(z)=z^2$ on $\mathbb{C}$.
   1. Determine the real and imaginary parts of $f$.
   2. Calculate the gradients of $u$ and $v$. (Recall that $$(\mathrm{\nabla }h)(x,y)=⟨({\partial }_{1}h)(x,y),({\partial }_{2}h)(x,y)⟩$$ is the gradient of $h:{\mathbb{R}}^2\to \mathbb{R}$.)
   3. Verify that $\mathrm{\nabla }u$ and $\mathrm{\nabla }v$ are perpendicular.
3. Let $f(z)=1/z$ on $\mathbb{C}\setminus \{0\}$.
   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 $\mathbb{C}\setminus \{0\}$.
5. For each of the following, determine the limit or explain why it does not exist.
   1. $\underset{z\to 0}{lim}{\displaystyle \frac{|z|}{z}}$
   2. $\underset{z\to 0}{lim}{\displaystyle \frac{|z{|}^2}{z}}$
   3. $\underset{z\to 0}{lim}\mathsf{Arg}(z)$
6. Prove that the following functions are continuous.
   1. $f(z)=\mathsf{Re}(z)$ on $\mathbb{C}$
   2. $f(z)=|z|$ on $\mathbb{C}$
   3. $f(z)=\mathsf{Arg}(z)$ on $\mathbb{C}\setminus (-\mathrm{\infty },0]$

Holomorphic Functions

7. Use the definition of the derivative to differentiate each of the following functions.
   1. $f(z)=z^2+z$
   2. $f(z)={\displaystyle \frac{1}{z}}$
8. Define $f:\mathbb{C}\to \mathbb{C}$ 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 $h:{\mathbb{R}}^2\to \mathbb{R}$ 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 $z\in \mathbb{C}$ where $f(z)=|z|$ is differentiable.
12. Verify that the functions $$u(x,y)=x^3-3x{y}^{2}v(x,y)=3x^{2}y-y^3$$ satisfy the Cauchy-Riemann equations for all $(x,y)$. Show that $u,v$ are the real and imaginary parts of a holomorphic function $f:\mathbb{C}\to \mathbb{C}$.
13. Verify that the functions $$u(x,y)=\frac{x^4-6x^2{y}^2+y^4}{(x^2+y^2{)}^4}v(x,y)=\frac{4x{y}^3-4x^{3}y}{(x^2+y^2{)}^4}$$ 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 $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$.

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 $${\partial }_1({\partial }_{1}u)+{\partial }_2({\partial }_{2}u)=0{\partial }_1({\partial }_{1}v)+{\partial }_2({\partial }_{2}v)=0$$ both hold.

    (The Laplacian of $g:{\mathbb{R}}^2\to \mathbb{R}$ is $\mathrm{△}g={\partial }_1({\partial }_{1}g)+{\partial }_2({\partial }_{2}g)$. Functions satisfying the Laplace equation $\mathrm{△}g=0$ 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 $\mathbb{C}$. Suppose we know that $u(x,y)=x^5-10x^3{y}^2+5x{y}^4$. 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 $$u(x,y)=x^3-kx{y}^2+12xy-12x$$ for some constant $k\in \mathbb{C}$. Find all values of $k$ for which $u$ is the real part of a holomorphic function $f:\mathbb{C}\to \mathbb{C}$.
18. Show that if $f:\mathbb{C}\to \mathbb{C}$ is holomorphic and $f$ has a constant real part then $f$ is constant.
19. Show that the only holomorphic function $f:\mathbb{C}\to \mathbb{C}$ of the form $f(x+iy)=u(x)+iv(y)$ is given by $f(z)=\lambda z+a$ for some $\lambda \in \mathbb{R}$ and $a\in \mathbb{C}$.
20. Suppose that $f(x+iy)=u(x,y)+iv(x,y)$. is a holomorphic function and that $$2u(x,y)+v(x,y)=5$$ for all $x+iy\in \mathbb{C}$. Prove that $f$ is constant.