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 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 $u=\cos^{2}3\theta$ 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 $u$.

Problem Set 3 (due 02/26/2018, Mon)

• Prob 1. Consider the function $\dfrac{1}{z(3-z)}.$
  • 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 03/05/2018, Mon)

• Prob 1. Evaluate the following integrals:
  • $\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
  • $\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
• 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 $n.$

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 single-valued in the complex plane.

Problem Set 7 (due 04/04/2018, Mon)

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

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

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

• 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 $u(x,a)=1/(1+x^{2}).$ The solution is assumed to vanish at the infinities of $x$. Find $u(x,y).$

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

Problem Set 12 (do not hand in)

• Prob 1.

  • Find the Green's function which satisfies the partial differential equation  
    $(i\dfrac{\partial}{\partial t}+\dfrac{\partial^{2}}{\partial x^{2}})G(x-x^{\prime},t-t^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime}),$ $\ -\infty$ < $x$ < $\infty,$

    the initial condition

    $G(x-x^{\prime},t-t^{\prime})=0,$ $t$ < $t^{\prime},$

    and the boundary condition

    $G(\pm\infty,t-t^{\prime})=0.$

  • 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

    $\Psi(\pm\infty,t)=0$.