Problem Sets

Problem Set 1 (due 02/11/2019, M)

Prob 1. Find the general solution of $y^{\prime\prime}-y=x^{2}+\cos2x+e^{4x}.$ Determine the solution of these equations satisfying $y(0)=0,$ $y^{\prime}(0)=1.$

Problem Set 2 (due 02/19/2019, T)

Prob 1. Chapter 1, 1a, 1c, 2f.

Also, find the general solution of

$ (D+1)^{3}y=e^{-x}.$
Prob 2.
- Find the real part and the imaginary part of $i^{3i}.$ (Hint: there may be more than one set of such values.)
- Find the roots of $\ (z+1)^{4}=(z^{2}-1)^{4}.$

Problem Set 3 (due 02/25/2019, M)

Prob 1.
- Is there an analytic function of which the imaginary part is $xe^{y}$?
- What is $a$ if $u=x^{4}+ax^{2}y^{2}+y^{4}$ is the real part of an analytic function? Find the imaginary part of this analytic function. Express this analytic function as a function of $z$.
Prob 2. Let $u$ satisfies the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$ where $R$ is the disk $x^{2}+y^{2} $ < $9$ and $u$ satisfies the boundary condition $u=\cos^{2}3\theta$ at $r=3$. Find $u$.

Problem Set 4 (due 03/04/2019, M)

Prob 1. Give the explicit form for the divergence theorem in two dimensions. Prove its validity for a rectangle.

Prob 2. Consider the function $\dfrac{1}{z(5-z)}.$

Find its Taylor series expanded around $z=1.$ Where is this series convergent?
Find its various Laurent series expanded around $z=0$.

Prob 3. Evaluate the integral

$\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$

Problem Set 5 (due 03/11/2019, M)

Prob 1. Do Prob 2b of Problem set 4.

Prob 2.

Prove the Liouville theorem that if an entire function $f(z)$ is bounded at infinity, it is a constant.

Hint: show that the derivative of $f(z)$ is identically equal to zero.
Prove that if an entire function divided by $z^{n}$ is bounded at infinity, it is a polynomial of order no more than $n.$

Problem Set 6 (due 03/18/2019, Mon)

Prob 1. Evaluate the following integral:

$\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$

Prob 2. Calculate the integral $I=\int_{-\infty}^{\infty}\dfrac{x\sin x}{x^{2}}dx$ by deforming the contour away from the origin.

Prob 3. Problem 4a and 4b of chapter 2 in the text book.

Problem Set 7 (due 03/18/2019, Mon)

Prob 1. Calculate the integral $I=\int_{-\infty}^{\infty}\dfrac{x\sin x}{x^{2}}dx$ by replacing the integral with its principal value.

Prob 2. Calculate the integral $I=\int_{-\infty}^{\infty}\dfrac{1-\cos2\pi x}{(1-x^{2})}dx$ by whatever way you desire.

Problem Set 8 (due 04/08/2019, Mon)

Prob 1. Find the Fourier transform of $f(x)=e^{-x^{2}},-\infty$ < $x$ < $\infty.$

Prob 2. Let

$f(x)=e^{-x},$ $x>0,$

$=0,$ $x<0.$

Find $\widetilde{f}(k)$, the Fourier transform of $f(x).$
Apply the inversion formula

$f(x)=\int_{-\infty}^{\infty}\dfrac{dk}{2\pi}e^{ikx}\widetilde{f}(k)$

to determine $f(x)$ from the Fourier transform you found in (a).
Set $k=0$ in the inversion formula to find $f(0).$ Explain the result you get.

Prob 3. Find $u(r,\theta)$ which satisfies the two-dimensional Laplace equation

$\nabla^{2}u(r,\theta)=0$ , $r<1,$

as well as the boundary condition

$u(1,\theta)=\dfrac{1}{2+\cos\theta}.$

Problem Set 9 (due 04/17/2019, Wed)

Prob 1. Prove that, in the limit $\epsilon\rightarrow0,$ the Fourier transform of $f(x)=e^{-\epsilon\left\vert x\right\vert }$
$-\infty$ < $x$ < $\infty,$ is Dirac's delta function $\delta(k)$ $.$

Prob 2. Let $f(\theta)=\theta,$ $0<\theta<\pi$ and let the Fourier cosine series of $f(\theta)$ be

$f(\theta)=a_{0}+ \displaystyle\sum_{n=1}^{\infty} A_{n}\cos n\theta$.

Find $a_{0}$ and $A_{n}.$
By setting $\theta$ to zero in this series, prove that

$\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\cdot\cdot\cdot.$
Use the formula $\dfrac{1}{\pi}\int_{0}^{\pi}f^{2}(\theta)d\theta = a_{0}^{2}$ + $\dfrac{1}{2} \displaystyle\sum_{n=1}^{\infty} A_{n}^{2}$ to prove that

$\dfrac{\pi^{4}}{96}=1+\dfrac{1}{3^{4}}+\dfrac{1}{5^{4}}+\cdot\cdot\cdot.$

Prob 3. Find $u(r,\theta)$ which satisfies the two-dimensional Laplace equation

$\nabla^{2}u(r,\theta)=0$ , $r<1,$

as well as the boundary condition

$u(1,\theta)=\dfrac{1}{2+\cos\theta}.$

Problem Set 10 (due 04/29/2019, Mon)

Prob 1. Let $f(t)=t^{n},0$ < $t$ < $\infty,$

where $n$ is a non-negative integer.

Find the Laplace transform of $f(t).$
Verify for the present example the validity of the inversion formula for Laplace transform.

Prob 2. Let the solution of the differential equation

$\dfrac{d^{2}x}{dt^{2}}-4x=\cos t,$ $t>0$

satisfy the initial conditions

$x(0)=A,$ $\overset{\cdot}{x}(0)=B.$

Find this solution with the use of Laplace transform\bigskip.
Find the general solution of the differential equation with the operator method you learned earlier in the semester. Determine the arbitrary constants in this solution with the initial conditions.
Which of these two methods do you consider the more efficient? Why?

Problem Set 11 (due 05/06/2019, Mon)

Prob 1. Consider the 2-D Laplace equation $\nabla^{2}u=0$ which holds inside the
infinite strip $0$ < $y$ < $a,$ $-\infty$ < $x$ < $\infty.$ The boundary conditions for $u(x,y)$ are $u(x,0)=0$ and $u(x,a)=1/(1+x^{2}).$ The solution is assumed to vanish at the infinities of $x$. Find $u(x,y).$

Prob 2. Consider the Schrödinger equation in a one-dimensional space

$i\dfrac{\partial\Psi}{\partial t}=-\dfrac{\partial^{2}\Psi}{\partial x^{2}},$ $\Psi=\Psi(x,t),$

which holds for all values of $x$ and all positive values of $t$. Let the initial value of $\Psi(x,t)$ be

$\Psi(x,0)=f(x),$

and let $\Psi$ vanish at the infinities of space.

Find the equation satisfied by $\overset{\sim}{\Psi}(k,t)$ (the Fourier transform of $\Psi(x,t)\ $with respect to $x)$ and solve the resulting equation.
Express the solution $\Psi(x,t)$ satisfying the initial condition in the form of a Fourier integral.
Find the Green function $G(x-x^{\prime},t)$ in closed form for the Schrödinger equation so that the solution in (b) can be expressed as

$\Psi(x,t)=\int_{-\infty}^{\infty}G(x-x^{\prime},t)f(x^{\prime})dx^{\prime}$.