How to do integrals involving Gauss function and cosine function?

Question

How can I calculate the integral: $$\int_{-\infty}^{+\infty}e^{-\frac{(x-a)^2}{0.01}}\cos(bx)dx$$ I can not find in the references. Excuse my bad English.

Answer 1

With

$$ \cos(bx)=\frac12\left(\mathrm e^{\mathrm ibx}+\mathrm e^{-\mathrm ibx}\right)\;, $$

the integrand becomes the sum of two Gaussians with complex exponents, whose integrals can be evaluated like this.

Answer 2

A reference is Rudin, Principles of mathematical analysis, Example 9.43. Here the integral $$ \int_{-\infty}^{\infty}e^{-x^2}\cos(xt)\,dx $$ is calculated using the theory of ordinary differential equations. The integral is $$ \sqrt{\pi}\exp\left(-\frac{t^2}{4} \right). $$ (Hint) In your integral after introducing new variable you should calculate $$ \int_{-\infty}^{\infty}e^{-z^2}\cos(cz)\,dz $$ and $$ \int_{-\infty}^{\infty}e^{-z^2}\sin(cz)\,dz. $$