MathDB

The integral

Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}

(a)(3 POINTS:)Show that the integral

Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}

Where

j

is not a function of

x

,is

Z(j)=e^{j^{2}/4a} Z(0)

(b)(10 POINTS):Show that,

\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}

Where

(2n-1)!!

is defined as

(2n-1)(2n-3)\times...\times3\times 1

(c)(7 POINTS):What is the number of ways to form

n

pairs from

2n

distinct objects?Interept the previous part of the problem in term of this answer.

Problem Statement