18.075 - Advanced Calculus for Engineers (Spring 2023)

Problem Sets

Problem Set 1 (due 02/13/2023, M)

• Prob 1. Find the general solution of the differential equation

  $y^{\prime\prime}-y=x^{2}+\sin3x+e^{x}+e^{\sin x}.$

• Prob 2. Prob 1a and 1c in Chapter 1.

Problem Set 2 (due 02/21/2023, Tu)

• Prob 1. Prob 1 in Chapter 2.
• Prob 2. Prob 2 in Chapter 2.
• Prob 3. Find the real part and the imaginary part of $\cos z$, where $z=x+iy.$

Problem Set 3 (due 03/06/2023, Mon)

• Prob 1. Let $u$ satisfies the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ in a region $R$ where $R$ is the disk $x^{2}+y^{2}$ < $9$ and $u$ satisfies the boundary condition $u=\cos^{2}4\theta$ at $r=3$. Find $u$

• Prob 2.

  1. 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.

  2. Prove that if an entire function divided by $z^{n}$ is bounded at infinity, it is a polynomial of order no more than $n.$

• Prob 3 Consider the function $\dfrac{1}{z(9-z)}.$

  1. Find its Taylor series expanded about $z=1.$ Where is this series convergent?

  2. Find its various Laurent series expanded about $z=0$.

Problem Set 4 (due 03/20/2023, Mon)

• Prob 1. Consider the function $\dfrac{1}{z(9-z)}.$

  1. Find its Taylor series expanded about $z=1.$ Where is this series convergent?

  2. Find its various Laurent series expanded about $z=0$.

• Prob 2. Problem 4a and 4d in Chapter 2 of the text book.

• Prob 3. Problem 4b and 4e in Chapter 2 of the text book.

Problem Set 5 (due 04/10/2023, Mon)

• Prob 1. Find the Fourier transform of $f(x)=e^{-ax^{2}},-\infty$ < $x$ < $\infty,$ where $a$ is a positive constant.$.$

• Prob 2.The Dirac Delta function $\delta(x)$ is defined to be

  $\delta(x)=0,$ $x\neq0$

  and

  $\int_{-\infty}^{\infty}\delta(x)$ $f(x)$ $dx=f(0).$

  This function is conveniently expressed as

  $\delta(x)=\int_{-\infty}^{\infty}\dfrac{dk}{2\pi}e^{ikx},$

  which is not a convergent integral. Prove that a more rigorous definition for the Dirac Delta function is

  $\delta(x)=\underset{\epsilon\longrightarrow0}{\lim}\int_{-\infty}^{\infty}\dfrac{dk}{2\pi}e^{ikx}e^{-\epsilon x^{2}}.$

• Prob 3. Let

  $f(x)=e^{-2x},$     $x>0,$

       $=0,$       $x$ < $0.$

  1. Find $\widetilde{f}(k)$, the Fourier transform of $f(x).$

  2. Apply the inversion formula

     $f(x)=\int_{-\infty}^{\infty}\dfrac{dk}{2\pi}e^{ikx}\widetilde{f}(k)$

     to determine $f(x)$ from the Fourier transform you found in (a).

  3. Set $k=0$ in the inversion formula to find $f(0).$ Explain the result you get.

Problem Set 6 (due 04/24/2023, Mon)

• Prob 1. Consider the heat equation with a heat source $\rho(x,t)$:

  $\dfrac{\partial\Psi(x,t)}{\partial t}=\dfrac{\partial^{2}\Psi(x.t)}{\partial x^{2}}+\rho(x,t).$

  The initial condition is

  $\Psi(x.0)=f(x),$

  and the boundary conditions are

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

  1. Find the complementary solution of this problem.

  2. Find the particular solution of this problem satisfying

     $\Psi_{p}(x.0)=0.$

     Use the convolution theorem to express $\Psi_{p}(x.t)$ as an integral over the product of a Green's function and $\rho(x,t)$.

• Prob 2. Let $G(x-x^{\prime},t-t^{\prime})$ satisfy the equation

  $(\dfrac{\partial}{\partial t}-\dfrac{\partial^{2}}{\partial x^{2}})G(x-x^{\prime},t-t^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime})$

  and the condition of causality

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

  1. Find $G(x-x^{\prime},t-t^{\prime})$.

  2. Show that $G(x-x^{\prime},t-t^{\prime})$ is the same Green's function found in Prob. 1(b)