Fourier transform of integral e^-a|x|

[RedDwarf]

Homework Statement

I am supposed to compute the Fourier transform of f(x) = integral (e^-a|x|)

Homework Equations

Fourier transformation:
F(p) = 1/(2π) ^n/2 integral(f(x) e^-ipx dx) from -infinity to +infinity

The Attempt at a Solution

My problem is, that I do not know how to handle that there are no boundaries in the integral. If there were from minus to plus infinity it would be okay to solve (but then the transform would be irrational).
With the boundaries I would split the integral for the negative and positive x and get f(x)=2/a and therefore constant.
The Fourier transform would then only consist of a constant times integral(e ^-ipx) . As this integral only gives an infinite solution, I am rather confused.
Thanks for help!

 

[Orodruin]

I would say that f(x) is likely not supposed to be a definite integral, but a primitive function of $e^{-a|x|}$. Either way, this is impossible to know without asking the person who constructed the problem.

 

[RedDwarf]

I would say that f(x) is likely not supposed to be a definite integral, but a primitive function of $e^{-a|x|}$. Either way, this is impossible to know without asking the person who constructed the problem.

This is what I think as well. If they meant the primitive function though, how could I solve it then?

 

[Orodruin]

What is the relationship between a function $f(x)$ and its primitive function $F(x)$?

 

[RedDwarf]

What is the relationship between a function $f(x)$ and its primitive function $F(x)$?

The function is the derivative of the primitive function $\frac{d}{dx} F(x) = f(x)$
Yet, I am not sure how this is helping me in this Fourier transform (sorry, I'm a bit slow today)

 

[Orodruin]

What can you say about the Fourier transform of a general function g(x) and its relation to the Fourier transform of its derivative g’(x)?

 

[RedDwarf]

What can you say about the Fourier transform of a general function g(x) and its relation to the Fourier transform of its derivative g’(x)?

Ah, now I know where this is going, thanks!
The Fourier transform of the derivative of a general function is related to the function like so: $\hat{g'}(x) = ip \hat{g}(x)$.
In my case this would mean that I can look at the Fourier transform of the derivative, divided by ip:
$\hat{f}(x) = \frac{1}{ip} \hat{f'} = \frac{1}{i\sqrt{2p^2 \pi}} \int e^{-a|x|} e^{-ipx} dx = Const. \cdot \int_{-\infty}^{\infty} e^{-a|x|-ipx} dx$
I can split up that last integral (in order to get rid of that absolute value of x):
$\int_{-\infty}^{\infty} e^{-a|x|-ipx} dx = \int_{-\infty}^{0} e^{x(a-ip)} dx + \int_{0}^{\infty} e^{x(-a-ip)} dx = \frac{1}/{a-ip} + \frac{1}{a+ip} = \frac{2a}{a^2+p^2}$
Combined with the constant from earlier:
$\hat{f}(x) = \frac{2a}{i p \sqrt{2\pi} (a^2+b^2)}$
That should be it. Thank you so much!