Problem Sets
Problem Set 1 (due 02/12/2018, M)
 Prob 1. Find the general solution of $y^{\prime\prime}y=x^{2}+\cos2x+e^{3x}.$ Determine the solution of these equations satisfying $y(0)=0,$ $y^{\prime}(0)=1.$
Problem Set 2 (due 02/20/2018, T)
 Prob 1. Chapter 1, 1a,1c,2f.
 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 $\ (z+1)^{4}=(z^{2}1)^{4}.$
 Prob 3.
 Is there an analytic function which has $xe^{y}$ as its imaginary part?
 What is $a$ if $u=xy^{3}+ax^{3}y$ 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 4. Let $u$ satisfies the twodimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$.
 If $R$ is the disk $x^{2}+y^{2}$<$9$ and $u$ satisfies the boundary condition $u=\cos^{2}3\theta$ at $r=3$.
Find $u$.  If $R$ is the annulus between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1,$ and $u$ satisfies the boundary condition $u=\cos2\theta$ at $r=2$ and $u=1$ at $r=1$. Find $u$.
 If $R$ is the disk $x^{2}+y^{2}$<$9$ and $u$ satisfies the boundary condition $u=\cos^{2}3\theta$ at $r=3$.
Problem Set 3 (due 02/26/2018, Mon)
 Prob 1. Consider the function $\dfrac{1}{z(3z)}.$
 Find its Taylor series expanded around $z=1.$ Where is this series convergent?
 Find its various Laurent series expanded around $z=0$.
Problem Set 4 (due 03/05/2018, Mon)
 Prob 1. Evaluate the following integrals:
 $\int_{\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x3i)(x5i)}.$
 $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
 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 $n.$
 Prove the Liouville theorem that if an entire function $f(z)$ is bounded at infinity, it is a constant.
Problem Set 5 (due 03/14/2018, Wed)
 Prob 1. Prove the fundamental theorem of algebra which says that if $P(z)$ is a polynomial of
order $n$, it has $n$ roots.
Hint: Apply Louiville's theorem to the inverse of $P(z).$  Prob 2. Prove that, if $R(z)$ is a rational function of $z$ vanishing at infinity faster than $z^{2},$ the sum of its residues in the complete $z$plane is equal to zero.
Hint: Calculate $\int_{\infty}^{\infty}R(x)dx$ in two ways with Cauchy's residue theorem. How do you modify your proof if $R(z)$ has poles on the real axis?
Problem set 6 (due 03/19/2018, Mon)
 Prob 1. Calculate the integral $I=\int_{\infty}^{\infty}\dfrac{x\sin x} {x^{2}}dx$ by deforming the contour away from the origin.
 Prob 2. Calculate $I\equiv\int_{\infty}^{\infty}\dfrac{x^{2}}{\cosh\pi x}dx$.
Prob 3. Find the branch points of the function
$f(z)=(z^{3}1)^{1/3}.$
 Is the infinity a branch point of $f(z)$? Draw a set of finite branch cuts for $f(z)$ to make this function singlevalued in the complex plane.
Problem Set 7 (due 04/04/2018, Mon)
 Prob 1. Problem 11a in Chapter 2.
 Prob 2. Calculate $I=\int_{0}^{1}(\dfrac{x}{1x})^{1/3}\dfrac{1}{1+x^{2}}dx.$
 Prob 3. Find the Fourier transform of 1/${\cosh x}.$ Show explicitly that Fourier's inversion formula is valid for this example.
Problem Set 8 (due 04/09/2018, Mon)
 Prob 1. Find the Fourier transform of $f(x)=e^{x^{2}},\infty $<$ x $<$\infty.$
Prob 2. Let $f(\theta)=\sin\theta,$ $0$ < $\theta $<$ \pi.$
Express $f(\theta)$ as a Fourier cosine series.
 Prob 3. 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, prove that the series above gives
$\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\cdot\cdot\cdot.$

From the formula
$\dfrac{1}{\pi}\int_{0}^{\pi}f^{2}(\theta)d\theta=a_{0}^{2}$ $+\dfrac{1}{2} \displaystyle\sum_{n=1}^{\infty} A_{n}^{2}$ $,$
prove that
$\dfrac{\pi^{4}}{96}=1+\dfrac{1}{3^{4}}+\dfrac{1}{5^{4}}+\cdot\cdot\cdot.$
Compare how rapidly the two series in (b) and (c) converge.
Problem Set 9 (due 04/18/2018, Wed)

Prob 1. Find the Fourier coefficients $a_{n}$ for
$f(\theta)=\dfrac{1}{2+\cos\theta},$ $\pi<\theta<\pi,$
where $a_{n}\equiv\int_{\pi}^{\pi}e^{in\theta}f(\theta)\dfrac{d\theta}{2\pi}.$
How do you explain the behavior of $a_{n}$ when $\left\vert n\right\vert>>1?$
Problem Set 10 (due 04/23/2018, Mon)

Prob 1. 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
 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 04/30/2018, Mon)

Prob 1. Apply the convolution theorem for Laplace transform to solve the integral equation
$x=\int_{0}^{x}\dfrac{1}{\sqrt{xx^{\prime}}}y(x^{\prime})dx^{\prime},$ $x>0.$

Prob 2. Consider the 2D 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 3. Consider the SchrÃ¶dinger equation in a onedimensional 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 $G(xx^{\prime},t)$ in closed form for the SchrÃ¶dinger equation so that the solution in (b) can be expressed as
$\Psi(x,t)=\int_{\infty}^{\infty}G(xx^{\prime},t)f(x^{\prime})dx^{\prime}$.
Problem Set 12 (do not hand in)
 Prob 1.

Find the Green's function which satisfies the partial differential equation
$(i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})G(xx^{\prime},tt^{\prime})=\delta(xx^{\prime})\delta(tt^{\prime}),$ $\ \infty$ < $x$ < $\infty,$the initial condition
$G(xx^{\prime},tt^{\prime})=0,$ $t$ < $t^{\prime},$
and the boundary condition
$G(\pm\infty,tt^{\prime})=0.$

Use the Green's function in (a) to solve the equation
$i\dfrac{\partial\Psi(x,t)}{\partial t}+\dfrac{\partial^{2}\Psi(x,t)}{\partial x^{2}}=\rho(x,t),$ $\infty$ < $x<\infty,$ $t>0,$
with the initial value
$\Psi(x,0)=f(x),$
and the boundary condition
$\Psi(\pm\infty,t)=0$.
