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

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Homework Help Overview

The discussion revolves around finding the Fourier transform of the function exp(-|x|). Participants are exploring the implications of the absolute value in the function and how it affects the integration limits.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are considering whether to split the function into two parts based on the sign of x, questioning the need for different limits of integration. There is also discussion about the behavior of the function in different intervals and the implications of assuming exp(-|x|) behaves like exp(-x) across the entire domain.

Discussion Status

The discussion is active, with participants providing clarifications and exploring different interpretations of the absolute value function. Some guidance has been offered regarding the necessity of breaking the integration into two parts, but no consensus has been reached on the approach to take.

Contextual Notes

There is some confusion regarding the definitions and properties of the absolute value function, particularly in relation to negative values of x. Participants are navigating these assumptions as they work through the problem.

samdawy
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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

 
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samdawy said:
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 :smile:...what do you get when you d that?
 
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?
 
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.
 
should not |x| for any negative value equal to +x ?

sorry but I am a little bit confused,
 
if x is negative, then +x is also negative isn't it?:wink:

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?
 
Yah, I got it

I really thank you for you clarification
 

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