1
$\begingroup$

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!

$\endgroup$
7
  • 1
    $\begingroup$ is there anything you want from us? $\endgroup$
    – V-X
    May 18, 2013 at 10:48
  • $\begingroup$ try to use function Ei, if you want some help... $\endgroup$
    – V-X
    May 18, 2013 at 10:50
  • $\begingroup$ One can use the same idea as in your previous question, except that one has to integrate w.r.t. parameter instead of differentiating. $\endgroup$ May 18, 2013 at 10:50
  • $\begingroup$ ok..thanks..I tried to solve the second case in the previous question and it turns to be equal to the first case. Is that correct? Now I'll try to do the 0-th moment. $\endgroup$
    – JFNJr
    May 18, 2013 at 10:56
  • 1
    $\begingroup$ If you need only the answer, I can calculate it with help of Wolfram Mathematica. $\endgroup$ May 18, 2013 at 15:22

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .