18.04 - Complex Variables with Applications (Fall 2023)

Problem Sets

When Submitting:

• Name your file 18.04_Pset_#_FirstName_LastName
• Put 18.04 Pset # in the subject of your email

Problem Set 1 (due 09/18/2023, Mon)

• Prob 1. Find the general solution for

  1. $\dfrac{d^{4}y}{dx^{4}}-y=\sin 2x+2x+e^{3x}.$

  2. $\dfrac{d^{2}y}{dx^{2}}-y=2+x^{2}+e^{x}.$

• Prob 2. Find the general solution of $\dfrac{d^{2}y}{dx^{2}}-y=0.$

  Determine the arbitrary constants in this solution if $y(0)=0,$ $y^{\prime }(0)=1.$

• Prob 3. Chapter 1, Prob 1a,1b.

Problem Set 2 (due 10/02/2023, Mon)

• Prob 1.

  1. Find the values of $i^{2i}.$

  2. Find the roots of $\ (z+1)^{4}=(z^{2}-1)^{4}.$

• Prob 2. 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 $z$.

Problem Set 3 (due 10/30/2023, Mon)

• Prob 1. Find the various Laurent series for $\dfrac{1}{z(z-1)}$expanded about the origin. Indicate the regions where each of these series are convergent.

• Prob 2. Evaluate the following integrals with the use of the Cauchy residue theorem:

  1. $\int_{-\infty }^{\infty }\dfrac{1}{x^{4}+1}dx$,

  2. $\int_{-\infty }^{\infty }\dfrac{1}{(x-i)(x-2i)(x-3i)(x+i)}dx$

  3. $\int_{0}^{2\pi }\dfrac{1}{5+\cos \theta }d\theta .$

• Prob 3. Evaluate the following integrals with the use of the Cauchy residue theorem:

  1. $\int_{0}^{\infty }\dfrac{\cos x}{1+x^{2}}dx,$

  2. $\int_{0}^{\infty }\dfrac{x\sin x}{1+x^{2}}dx,$

  3. $\int_{-\infty }^{\infty }\dfrac{\sin ^{2}x}{x^{2}}dx,$

Problem Set 4 (due 11/20/23, Mon)

• Prob 1. Find the Fourier transform of the Gaussian $f(x)=e^{-x^{2}},-\infty $ < $x$ < $ \infty .$ Show explicitly that Fourier's inversion formula is valid for this function.

• Prob 2. Let $u=u(x,y)$ satisfies $u_{xx}+u_{yy}=0$ in the upper plane $y$ > $0$ and $-\infty $ < $x$ < $\infty .$ The boundary conditions are
  $u(x,0)=0,$     $-0$ < $x$ < $\infty ,$
              $=1,$     $-1$ < $x$ < $0,$
              $=0,$     $-\infty$ < $x$ < $-1.$
  

  It is assumed that $u$ vanishes at infinity. Find $u.$

Problem Set 5 (due 12/08/2023, Fri)

• Prob 1. Let $f(\theta )=\theta ,$ $0$ < $\theta$ < $\pi .$

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

     $f(\theta)=a_{0}+\underset{n=1}{\overset{\infty}{\displaystyle\sum }}A_{n}\cos n\theta.$

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

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

  3. From the formula

     $\dfrac{1}{\pi}\int_{0}^{\pi}f^{2}(\theta)d\theta=a_{0}^{2}$ $+ \dfrac{1}{2}\underset{n=1}{\overset{\infty}{\displaystyle\sum }}A_{n}^{2}$ $,$

     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.

• Prob 2. Prob 13 in Chapter 2.

• Prob 3. 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 $\Psi (\pm \infty ,t)=0$.

  1. 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.

  2. Express the solution $\Psi(x,t)$ satisfying the initial condition in the form of a Fourier integral.

  3. 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

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