Dror Bar-Natan: Classes: 2003-04: Math 1350F - Knot Theory:

(42)

Next: Class Notes for Thursday November 13, 2003 Previous: Class Notes for Tuesday November 11, 2003

Homework Assignment 9: Gaussian integration in Rⁿ

Assigned Thursday November 13; due Thursday November 20 in class.

This document in PDF: HW.pdf

Important. No class next Tuesday (November 18th)! Thursday's class will meet as usual.

To be handed in.

1. For any even natural number $m$ compute the integral
   $$\int_{-\infty}^\infty dx x^me^{-x^2/2}.$$

2. For any odd natural number $m$ compute the integral
   $$\int_0^\infty dx x^me^{-x^2/2}.$$

3. For any natural number $n$ use Gaussian integration in ${\mathbb{R}}^n$ to compute the volume (area) of the $(n-1)$-dimensional sphere $S^{n-1}=\{x\in{\mathbb{R}}^n: \vert x\vert=1\}$ and the volume of the $n$-dimensional ball $B^n=\{x\in{\mathbb{R}}^n: \vert x\vert\leq 1\}$.

The generation of this document was assisted by L^ATEX2HTML.