BD-BDMO 2016 Problem 9

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$ . (a) 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) Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$ .(c) What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.