Calculate : $\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}\cos(bx)dx $

Question

How I calculate this integral: $$\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}\cos(bx)dx$$

where $a, b, \omega$ are constant.

Answer 1

As reuns commented, consider $$I=\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}\cos(bx)\,dx$$ $$J=\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}\sin(bx)\,dx$$ $$K=I+iJ=\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}e^{ibx}\,dx=\int_{-\infty}^{+\infty}e^{-(ax^2-i(\omega+b) x)}\,dx$$ $$L=I-iJ=\int_{-\infty}^{+\infty}e^{-ax^2}e^{i\omega x}e^{-ibx}\,dx=\int_{-\infty}^{+\infty}e^{-(ax^2-i(\omega-b) x)}\,dx$$ Now, for each $K,L$ integrals, complete the square and you will arrive to some gaussian integral eaxy to evaluate. Finally, $I=\frac 12(K+L)$.