Solutions

PROBLEM SETS

Prob 1. Chapter 1, 1a, 1c, 2f.
Prob 2. Find the general solution of $y^{\prime\prime}-y=x^{2}+\cos2x+e^{x}.$ Determine the solution of this equation satisfying $y(0)=0,$ $y^{\prime}(0)=1.$
Prob 3. Transform the nonlinear differential equation $y^{\prime}=x-y^{2}$ into a linear one. Do you recognize the resulting equation?

Prob1.
- Find the value of $i^{2i}.$
- Find the roots of $\ (z+1)^{4}=(z^{2}-1)^{4}.$
Prob 2. Find the real part and the imaginary part of $e^{iz}$ as well as those of $\cos z$. For what values of $z$ is the real part of $e^{iz}$ equal to $\cos z$?
Prob 3. For what constant value (or values ) of $a$ is the function $u=xy^{2}+ax^{3}$ 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 1. Let $u$ satisfies the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$.
- Let $R$ be the disk $x^{2}+y^{2}$ < $9$ and $u$ satisfies the boundary condition $u=\cos^{2}4\theta$ at $r=3$. Find $u$.
- Let $R$ be the annulus between the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}=1/4,$ and $u$ satisfies the boundary condition $u=\cos3\theta$ at $r=2$ and $u=1$ at $r=1/2$. Find $u$.
Prob 2. Prob 4a and 4d of Chapter 2.

Prob 1.
- Find the Taylor series expansion of $\dfrac{1}{z(1-z)}$ around $z=2.$ In which region is this series convergent?
- Find the Laurent series of $\dfrac{1}{z(1-z)}$ expanded around $z=2$ in other regions in the complex plane.
Prob 2. Problems 4b and 4e of Chapter 2.
Prob 3. Evaluate the integral $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+1)^{3}}.$

Prob 1. Calculate the integral $I=\int_{-\infty}^{\infty}\dfrac{\sin^{3}x}{x^{3}}dx$ in two ways:
- By deforming the contour away from the origin.
- By considering the principal value of $I$.
Prob 2. Calculate $I\equiv\int_{-\infty}^{\infty}\dfrac{x^{2}}{\cosh\pi x}dx$.
Prob 3. Find the branch points of the function
$f(z)=(z^{4}-1)^{1/4}.$
Is the infinity a branch point of $f(z)$? Draw a set of branch cuts for $f(z)$ to make this function single-valued in the complex plane.

Prob 1. Determine the number of zeroes of
- $z^{7}+3z+1$ in the first quadrant.
- $e^{z}+4z-1$ inside the circle $\left\vert z\right\vert$
Prob 2. The fundamental theorem of algebra says that a polynomial of the $n$-th order has $n$ roots in the complex plane. Prove it with the application of the Rouche's theorem.
Prob 3. Find the Fourier transform of 1/$\cosh x.$ Show explicitly that Fourier's inversion formula is valid for this example.

Prob 1. Let the solution of the differential equation

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

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?
Prob 2. Find the Fourier transform of the Markus integral

$I(x)=\int_{-\infty}^{\infty}e^{-\lambda(x-y)^{2}}\dfrac{dy}{1+e^{y}},$ where $\lambda$ is a constant.

Express $I(x)$ as its Fourier integral.
Prob 3. Problem 18 in Chapter 2.

Prob 1. Apply the convolution theorem for Laplace transform to solve the integral equation

$x=\int_{0}^{x}\dfrac{1}{\sqrt{x-x^{\prime}}}y(x^{\prime})dx^{\prime},$ $x>0.$
Prob 2. Find the solution $u(x,y)$ of

$u_{x}+(x^{3}+y)u_{y}=0$, $u(0,y)=\dfrac{1}{1+y^{2}}.$

Prob 1. The function $u(x,t)$ satisfies the wave equation $\dfrac{\partial^{2}u}{\partial x^{2}}=\dfrac{\partial^{2}u}{\partial t^{2}},$ $-\infty$ < $x$ < $\infty,$ $t>0.$ In addition, $u(x,0)=f(x)$, $u_{t}(x,0)=0.$
- Take the Laplace transform of the equation and find the equation satisified by $L(x,s)$, where
  
  $L(x,s)\equiv\int_{0}^{\infty}dt$ $e^{-st}u(x,t).$
- Assuming that both $f(x)$ and $L(x,s)$ have Fourier transforms, find $L(x,s)$ in the form of a Fourier integral. (You are allowed to differentiate a Fourier integral by differentiating its integrand.)
- Find $u(x,t).$
Prob 2. 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 $u(x,a)=1/(1+x^{2}).$ Periodic boundary conditions are imposed at the infinities of $x$. Find $u(x,y).$

Prob 1. Consider the Schr$\overset{\cdot \cdot }{o}$dinger 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 $ satisfy periodic boundary conditions 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 Schr$ \overset{..}{o}$dinger equation so that the solution in (b) can be expressed as
  
  $\Psi (x,t)=\int_{-\infty }^{\infty }G(x-x^{\prime },t)f(x^{\prime})dx^{\prime }$.
Note: $\int_{-\infty }^{\infty }e^{-ik^{2}}dk=e^{-i\pi /4}\sqrt{\pi }.$
Prob 2. Find the Green function in closed from for the wave equation in one spatial dimention:

$ u_{tt}=u_{xx}.$
Prob 3. Find the Green function $G(\overrightarrow{x}-\overrightarrow {x_{0}})$ which satisfies

$(\dfrac{\partial^{2}}{\partial x^{2}}+\dfrac{\partial^{2}}{\partial y^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}-1)G(\overrightarrow{x}-\overrightarrow {x_{0}})=\delta^{(3)}(\overrightarrow{x}-\overrightarrow{x_{0}}),$

subject to the boundary condition that it vanishes at infinity.