Definite Integral of Exponential Function

[singular]

[SOLVED] Definite Integral of Exponential Function

Homework Statement

I have an integral that I need to solve for a quantum physics problem

\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx

How would I go about solving this thing?

 

[Rainbow Child]

Split it into two intervals, i.e. (-\infty,0),\,(0,\infty) and make a change or variables in the first one x\to-x

 

[singular]

Split it into two intervals, i.e. (-\infty,0),\,(0,\infty) and make a change or variables in the first one x\to-x

\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx

Split into two intervals
\int^{\infty}_{0}e^{-a|x| - ikx}dx + \int^{0}_{-\infty}e^{-a|x| - ikx}dx

Change of variables in the second term x to -x
\int^{\infty}_{0}e^{-a|x| - ikx}dx - \int^{0}_{\infty}e^{-a|x| + ikx}dx

\int^{\infty}_{0}e^{-a|x| - ikx}dx + \int^{\infty}_{0}e^{-a|x| + ikx}dx

Are these steps what you are talking about?
What would I do from here?

 

[Rainbow Child]

Since x\in(0,\infty)\Rightarrow |x|=x. Now combine the two integrals.

 

[singular]

Since x\in(0,\infty)\Rightarrow |x|=x. Now combine the two integrals.

Oh...duh...thank you