Using Parseval's Theorem to evaluate an integral

[LancsPhys14]

We have the integral [MATH]\int_{-\infty}^{\infty}\frac{4}{1+4\pi ^{2}t^{2}}sinc(4t)dt[/MATH] which we want to evaluate.

I know Parseval's theorem as [MATH]\int_{-\infty}^{\infty}f(t)\overline{g(t)}dt = \int_{-\infty}^{\infty}F(s)\overline{G(s)}ds[/MATH] where [MATH]F(s)[/MATH] and [MATH]G(s)[/MATH] represent the Fourier transform of [MATH]f(t)[/MATH] and [MATH]g(t)[/MATH] respectively. Here [MATH]\overline{g(t)}[/MATH] and [MATH]\overline{G(s)}[/MATH]are the complex conjugates of [MATH]g(t)[/MATH] and [MATH]G(s)[/MATH].

We define the Fourier transform as [MATH]F[f(x)](s)=F(s)=\int_{-\infty}^{\infty}f(x)e^{-2\pi six}dx[/MATH].

Splitting our integral up using Parseval's theorem we establish that:

[MATH]f(t)=\frac{4}{1+4\pi ^{2}t^{2}}[/MATH]
[MATH]g(t)=sinc(4t)[/MATH][MATH]\implies \overline{g(t)}=sinc(4t)[/MATH] Am I correct in stating this? As to my knowledge this function contains no complex component

[MATH]F[f(t)](s)=\int_{-\infty}^{\infty}\frac{4}{1+4\pi ^{2}t^{2}}e^{-2\pi sit}dt[/MATH]
I am unsure how to evaluate the above integral as according to sources such as WolframAlpha the integral does no converge and when done without limits it gives a function I am not familiar with called the exponential integral function.

Any help in completing this problem would be appreciated.

 

[Cubist]

I wonder if you're supposed to apply Parseval's theorem in reverse with this question -taking the inverse Fourier transform not the forward transform. Would that make sense in the context of the original question? I had a quick look at a formula sheet of Fourier transforms and it appears this might be a good approach. Note it has been many years since I've done work in this field so don't take this approach as "solid advice"!

 

[LancsPhys14]

I think I managed to solve it. I rewrote the integral as [MATH]\int_{-\infty}^{\infty}\frac{4}{1+4\pi^{2}t^{2}}sinc(4t)dt = 2\int_{-\infty}^{\infty}\frac{2}{1+(2\pi t)^{2}}sinc(4t)dt[/MATH]
[MATH]\implies F(s)=F[\frac{2}{1+(2\pi t)^{2}}](s) = e^{-|s|}[/MATH]
[MATH]\implies G(s)=F[sinc(4t)](s) = \frac{1}{4}\Pi(\frac{s}{4})[/MATH]
[MATH]\implies \int_{-\infty}^{\infty}\frac{4}{1+4\pi^{2}t^{2}}sinc(4t)dt = 2\int_{-\infty}^{\infty}e^{-|s|}\frac{1}{4}\Pi(\frac{s}{4})ds[/MATH]
[MATH]= 2\int_{-1}^{1}\frac{1}{4}e^{-|s|}ds = 4\int_{0}^{1}\frac{1}{4}e^{-s}ds = \int_{0}^{1}e^{-s}ds = 1-\frac{1}{e}[/MATH]

 

[Cubist]

[MATH] 1-\frac{1}{e}[/MATH]

I don't think that's quite right (but it is very close).
Check the limits on the integration as you eliminate the rectangular function. You're aiming for a result close to 0.8647

BTW: I think this method assumes a normalised sinc() function in the original question, sinc(x) = sin(pi*x)/(pi*x)