$0$-th moment of product of gaussian and sinc function

Question

I would like to calculate the following integrals:

1. $$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$

2. $$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x\pm\pi)}{\pi ax\pm\pi}\right)^2\quad \exp(-bx^2) \,dx$$

Thanks!

Answer 1

I have got the following answers with help of Wolfram Mathematica 8.0:

1. $\int_{-\infty}^{\infty} \left(\frac{sin(\pi a x)}{\pi a x}\right) e^{-b x^2} dx = \frac1{a^2 \cdot \pi^\frac{3}{2}} \cdot \left( -\sqrt{b} + \sqrt{b} \cdot e^{-\frac{a^2 \cdot \pi^2}{b}} + a \cdot \pi^\frac{3}{2} \cdot \operatorname{Erf}\left( \frac{\pi a}{\sqrt{b }} \right) \right)$ only if $\operatorname{Re}(b) > 0$ and $a \ge 0$, where $\operatorname{Erf}(x) = \frac{2}{\sqrt{\pi}} \cdot \int_0^x e^{-t^2} dt$
2. Unfortunately, Mathematica falls to evaluate the integral $\int_{-\infty}^{\infty} \left(\frac{sin(\pi a x)}{\pi a x + \pi}\right)^2 e^{-b x^2} dx$

Hope, it helps.