How can I solve this kind of integrals?

Question

I=$\int_{-\infty}^{+\infty} e^{-x^2/(2L^2)} e^{-ikx} cos(k_ox)dx$

I know a long technique to solve this integral but I want someone to please suggest me some easy methods (or any book) to solve this kind of integrals. Thank you in advance.

Answer 1

Note that we can evaluate the integral $J(a,b)$ given by

$$\begin{align} J&=\int_{-\infty}^\infty e^{-ax^2+ibx}\,dx\\\\ &=e^{-a b^2/4}\int_{-\infty}^\infty e^{-a(x+ib/2)^2}\,dx\\\\ &=e^{-ab^2/4}\int_{-\infty}^\infty e^{-ax^2}\,dx\\\\ &=\sqrt{\frac{\pi}{a}}e^{-ab^2/4} \end{align}$$

by completing the square, exploiting Cauchy's Integral Theorem, and evaluating a Gaussian Integral.

Using $\cos(k_0x)=\frac{e^{ik_0x}-e^{-ik_0x}}{2}$, we have

$$I=\frac12 J(1/L,k_0-k)+\frac12J(1/L,-(k+k_0))$$

since

$$I=\frac12\int_{-\infty}^\infty e^{-x^2/L+i(k_0-k)x}\,dx+\frac12\int_{-\infty}^\infty e^{-x^2/L-i(k_0+k)x}\,dx$$