Math 642-600 Current Assignment — Spring 2019

• Read section 6.5.1, 6.5.2 and 7.1.
• Do the following problems.
  1. Section 6.4: 3, 7 (assume $a$ and $k$ are positive), 9, 20, 25.

  2. Let $\xi \in \mathbb R$ and consider the integral $G(\xi)=\int_{-\infty}^\infty e^{-x^2}e^{i\xi x}dx$. Show that $G(\xi)= \sqrt{\pi}e^{-\xi^2/4}$, given that $G(0)=\sqrt{\pi}$.

  3. Show that $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt$ is analytic for $\text{Re}(z)>0$.

Updated 3/8/2019 (fjn)