1 #complex/qual/work

Is the following function continuous, differentiable, continuously differentiable? $$\begin{align*} f: {\mathbf{R}}^2 &\to {\mathbf{R}}\\ f(x, y) &= \begin{cases} {xy \over \sqrt{x^2 + y^2}} & (x, y) \neq (0, 0) \\ 0 & \text{else}. \end{cases} \end{align*}$$

? #complex/qual/work

Show that $f(z) = z^2$ is uniformly continuous in any open disk $|z| < R$, where $R>0$ is fixed, but it is not uniformly continuous on $\mathbb C$.

6 #complex/qual/work

Let $F:{\mathbf{R}}^2\to {\mathbf{R}}$ be continuously differentiable with $F(0, 0) = 0$ and ${\left\lVert {\nabla F(0, 0)} \right\rVert} < 1$.

Prove that there is some real number $r> 0$ such that ${\left\lvert {F(x, y)} \right\rvert} < r$ whenever ${\left\lVert {(x, y)} \right\rVert} < r$.

2 Multivariable derivatives #complex/qual/work

• Complete this definition: “$f: {\mathbf{R}}^n\to {\mathbf{R}}^m$ is real-differentiable a point $p\in {\mathbf{R}}^n$ iff there exists a linear transformation…”

• #complex/qual/work Give an example of a function $f:{\mathbf{R}}^2\to {\mathbf{R}}$ whose first-order partial derivatives exist everywhere but $f$ is not differentiable at $(0, 0)$.

• #complex/qual/work Give an example of a function $f: {\mathbf{R}}^2 \to {\mathbf{R}}$ which is real-differentiable everywhere but nowhere complex-differentiable.

Implicit/Inverse Function Theorems

3 #complex/qual/work

Let $f:{\mathbf{R}}^2\to {\mathbf{R}}$.

• #complex/qual/work Define in terms of linear transformations what it means for $f$ to be differentiable at a point $(a, b) \in {\mathbf{R}}^2$.

• #complex/qual/work State a version of the inverse function theorem in this setting.

• #complex/qual/work Identify ${\mathbf{R}}^2$ with ${\mathbf{C}}$ and give a necessary and sufficient condition for a real-differentiable function at $(a, b)$ to be complex differentiable at the point $a+ib$.

5 #complex/qual/work

Let $P = (1, 3) \in {\mathbf{R}}^2$ and define $$\begin{align*} f(s, t) \coloneqq ps^3 -6st + t^2 .\end{align*}$$

• State the conclusion of the implicit function theorem concerning $f(s, t) = 0$ when $f$ is considered a function ${\mathbf{R}}^2\to{\mathbf{R}}$.

• State the above conclusion when $f$ is considered a function ${\mathbf{C}}^2\to {\mathbf{C}}$.

• Use the implicit function theorem for a function ${\mathbf{R}}\times{\mathbf{R}}^2 \to {\mathbf{R}}^2$ to prove (b).

There are various approaches: using the definition of the complex derivative, the Cauchy-Riemann equations, considering total derivatives, etc.

7 #complex/qual/work

State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.

Complex Differentiability

4 #complex/qual/work

Let $f = u+iv$ be complex-differentiable with continuous partial derivatives at a point $z = re^{i\theta}$ with $r\neq 0$. Show that $$\begin{align*} {\frac{\partial u}{\partial r}\,} = {1\over r}{\frac{\partial v}{\partial \theta}\,} \qquad {\frac{\partial v}{\partial r}\,} = -{1\over r}{\frac{\partial u}{\partial \theta}\,} .\end{align*}$$

Tie’s Extra Questions: Fall 2016

Let $u(x,y)$ be harmonic and have continuous partial derivatives of order three in an open disc of radius $R>0$.

• Let two points $(a,b), (x,y)$ in this disk be given. Show that the following integral is independent of the path in this disk joining these points: $$\begin{align*} v(x,y) = \int_{a,b}^{x,y} ( -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy) .\end{align*}$$

• In parts:

• Prove that $u(x,y)+i v(x,y)$ is an analytic function in this disc.
• Prove that $v(x,y)$ is harmonic in this disc.

Tie’s Questions, Spring 2014: Polar Cauchy-Riemann #complex/qual/work

Let $f=u+iv$ be differentiable (i.e. $f'(z)$ exists) with continuous partial derivatives at a point $z=re^{i\theta}$, $r\not= 0$. Show that $$\begin{align*} \frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta} .\end{align*}$$

? #complex/qual/work

• Show that the function $u=u(x,y)$ given by $$\begin{align*} u(x,y)=\frac{e^{ny}-e^{-ny}}{2n^2}\sin nx\quad \text{for}\ n\in {\mathbf N} \end{align*}$$ is the solution on $D=\{(x,y)\ | x^2+y^2<1\}$ of the Cauchy problem for the Laplace equation $$\begin{align*}\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0,\quad u(x,0)=0,\quad \frac{\partial u}{\partial y}(x,0)=\frac{\sin nx}{n}.\end{align*}$$

• Show that there exist points $(x,y)\in D$ such that $\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}$.