Continuity

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 \(\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}\).

#complex/qual/work