MATH529: Mathematical methods for the physical sciences IIMay 13, 2021

Solve the following problems (4 course points each). Present a brief motivation of your method of solution. Answers without explanation of solution procedure are not awarded credit.

1. Use the Laplace transform $F\left(s\right)=ℒ\left\{f\left(t\right)\right\}={\int }_0^{\infty }{e}^{-st}f\left(t\right)dt$ to solve the problem

   ${\displaystyle a^2\frac{{\partial }^{2}u}{\partial x^2}=\frac{{\partial }^{2}u}{\partial t^2},x>0,t>0,}$ ${\displaystyle u(0,t)=0,{lim}_{x\to \infty }\frac{\partial u}{\partial x}(x,t)=0,t>0,}$ ${\displaystyle u(x,0)=0,v(x,0)=\frac{\partial u}{\partial t}(x,0)=-v_0,x>0,v\in {ℝ}_{+}.}$

2. Use the Fourier transform $F\left(\alpha \right)=ℱ\left\{f\left(x\right)\right\}={\int }_{-\infty }^{\infty }{e}^{i\alpha x}f\left(x\right)dx$ to solve the problem

   ${\displaystyle \frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0,0<x<\pi ,y>0,}$ ${\displaystyle u(0,y)=f\left(y\right),\frac{\partial u}{\partial x}(\pi ,y)=0,y>0,}$ ${\displaystyle \frac{\partial u}{\partial y}(x,0)=0,0<x<\pi .}$

3. Find all solutions of the equation $z^8-2z^4+1=0.$ Write the roots in both Cartesian and polar form.

4. Sketch the region defined by $-1⩽\mathrm{Im}(1/z)<1$. Is this region a domain?

5. Is $f\left(z\right)=x^2-x+y+i(y^2-5y-x)$ an analytic function? Is it differentiable along the curve $y=x+2$?

6. Evaluate the integral

   ${\displaystyle \underset{C}{\oint }\frac{2z}{z^2+3}dz}$

   for $C$ defined as:

   1. $\left|z\right|=1$;

   2. $|z-2i|=1$.

7. Expand

   ${\displaystyle f\left(z\right)=\frac{1}{z{(1-z)}^2}}$

   in a Laurent series valid for:

   1. $0<\left|z\right|<1$;

   2. $\left|z\right|>1$.