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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!

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is there anything you want from us? –  V-X May 18 '13 at 10:48
    
try to use function Ei, if you want some help... –  V-X May 18 '13 at 10:50
    
One can use the same idea as in your previous question, except that one has to integrate w.r.t. parameter instead of differentiating. –  O.L. May 18 '13 at 10:50
    
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. –  JFNJr May 18 '13 at 10:56
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If you need only the answer, I can calculate it with help of Wolfram Mathematica. –  Piotr Semenov May 18 '13 at 15:22
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1 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.

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Thanks. It helps. –  JFNJr May 20 '13 at 13:52
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