How to find the fourier transform of exp(-|x|)

[samdawy]

Homework Statement

I have been trying to solve the Fourier transform of exp(-|x|)

Homework Equations

Do I need to split the function into two parts with different limits,i.e. the first has a limit from -infinity to zero and the secod from zero to +infinity. Please advise?

The Attempt at a Solution

 

[gabbagabbahey]

Do I need to split the function into two parts with different limits,i.e. the first has a limit from -infinity to zero and the secod from zero to +infinity. Please advise?

That sounds like a good plan to me ...what do you get when you d that?

 

[samdawy]

To be honest, I just assumed the exp(-|x|) is equal to exp(-x) for x between the minus and positive infinity and did the normal integration. Am I in the right track?

 

[gabbagabbahey]

Well, |x| is equal to -x for negative values of x isn't it?...And so in the interval -inf to 0, exp(-|x|)=exp(+x)...that is why you need to break the integration into two parts.

 

[samdawy]

should not |x| for any negative value equal to +x ?

sorry but I am a little bit confused,

 

[gabbagabbahey]

if x is negative, then +x is also negative isn't it?

For example, let's look at x=-2...clearly +x=-2 while -x=+2 so |x|=-x in this case since |-2|=2...do you follow?

 

[samdawy]

Yah, I got it

I really thank you for you clarification