Homework 1: Complex Analysis

Due February 7th

Problem 1

If $latex f$ is differentiable at $latex z$, then $latex f$ is continuous at $latex z$.

Problem 2

Let $latex T:\mathbb C \to \mathbb C$ be $latex \mathbb R$-linear.

1. $latex T$ is $latex \mathbb C$-linear if and only if $latex T(iz) = iTz$ for all $latex z\in\mathbb C$.
2. If $latex T$ has matrix $latex \begin{pmatrix}a & b\\c & d\end{pmatrix}$, then $latex T$ is $latex \mathbb C$-linear if and only if $latex a = d$ and $latex b = -c$.

Problem 3

Define $latex f:\mathbb C\to\mathbb C$ by

$latex f(x+iy) = \begin{cases} 0 & x=0\\ 0 & y=0\\ 1 & \text{otherwise.}\end{cases}$

Then $latex f$ satisfies the Cauchy-Riemann equations at the origin, but it is not differentiable at the origin.

Problem 4

Define $latex f:\mathbb C\to\mathbb C$ by

$latex f(z) = \begin{cases} |z|^2\sin(1/|z|) & z\not=0\\ 0 & z=0.\end{cases}$

Then $latex f$ is differentiable at the origin, but the partial derivative $latex \partial_x u$ is not continuous at the origin.

Problem 5

Let $latex a\in[0,1]$. Then there exist sequences $latex x_n, y_n$ of positive numbers converging to 0 such that $latex x_n^{y_n}\to a$.