Problem Sets

Problem Set 1 (due 02/15/2017, Wed)

• Prob 1. Chapter 1, 1a, 1c, 2f
• Prob 2. Find the general solution of $y^{\prime\prime}-y=x+\cos3x+e^{4x}.$ Determine the solution of this equation satisfying $y(0)=0,$ $y^{\prime}(0)=1.$

Problem Set 2 (due 03/01/2017, Wed)

• Prob 1.
  • Find the real part and the imaginary part of $i^{2i}.$ (Hint: there may be more than one set of such values.)
  • Find the roots of $\ (z+1)^{3}=(z^{2}-1)^{3}.$
• Prob 2.
  • Is $xy^{2}$ the imaginary part of an analytic function?
  • 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 3. Let $u$ satisfies the two-dimensional 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 $u=\cos^{2}5\theta$ 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 $u=\cos3\theta$ at $r=2$ and $u=1$ at $r=1$. Find $u$.

Problem Set 3 (due 03/08/2017, Wed)

• Prob 1. Evaluate the following integrals:
  • $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+4)(x-3i)(x-5i)}.$
  • $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\cos\theta)^{2}}.$
• Prob 2. Consider the function $\dfrac{1}{z(2-z)}.$
  • 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 03/15/2017, Wed)

• Prob 1. Evaluate the integral $\int_{-\infty}^{\infty}\dfrac {dx}{(x^{2}+1)^{4}}.$
• Prob 2. Evaluate the integral $\int_{-\infty}^{\infty}\dfrac{\cos^{2}(\pi x/2)}{(1-x^{2})}dx.$

Problem Set 5 (due 03/24/2017, Fri)

• Prob 1. Calculate the integral $I=\int_{-\infty}^{\infty}\dfrac{1-\cos 3x}{x^{2}}dx$ in two ways:
  • By deforming the contour away from the origin.
  • By considering the principal value of $I$.
• Prob 2. Calculate $I\equiv\int_{-\infty}^{\infty}\dfrac{x}{\sinh\pi x}dx$.

• 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 single-valued in the complex plane.

Problem Set 6 (due 04/10/2017, Mon)

• Prob 1. Problem 11a in Chapter 2.
• Prob 2. Calculate $I=\int_{0}^{1}(\dfrac{x}{1-x})^{1/3}\dfrac{1}{1+x^{2}}dx.$
• Prob 3. Find the Fourier transform of 1/$\cosh x.$ Show explicitly that Fourier's inversion formula is valid for this example.
• Prob 4. Let $f(\theta)=\theta,$ $0$ < $\theta$ < $\pi.$

  • Find the coefficients of $f(\theta)$ in the Fourier cosine series

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

  • By setting $\theta$ to zero, prove from the series in (a) that

    $\dfrac{\pi^{2}}{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\cdot\cdot\cdot.$

  • 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 7 (due 04/26/2017, Wed)

• 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?

• Prob 2. Apply the convolution theorem for Laplace transform to solve the integral equation

  $x=\int_{0}^{x}\dfrac{1}{\sqrt{x-x^{\prime}}}y(x^{\prime})dx^{\prime},$ $x>0.$

Problem Set 8 (due 05/08/2017, 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

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

    Note: $\int_{-\infty}^{\infty}e^{-ik^{2}}dk=e^{-i\pi/4}\sqrt{\pi}.$

• Prob 3. With the use of Fourier transform, find the solution of the Klein-Gordon equation

  $(\dfrac{\partial^{2}}{\partial t^{2}}-\dfrac{\partial^{2}}{\partial x^{2}}+m^{2})\phi(x,t)=0,$ $-\infty$ < $x$ < $\infty,$ $t>0,$ $m$ a constant,

  wth the initial conditions

  $\phi(x,0)=f(x),$ $\phi_{t}(x,0)=g(x).$

  You may assume that the function $\phi(x,t)$ vanishes at the spatial infinity. Express $\phi(x,t)$ as a single integral over the spatial variable with its integrand involving $f(x)$ and $g(x)$. Identify the Green functions in your expression.