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Definite Integral of Exponential Function

  1. Feb 19, 2008 #1
    [SOLVED] Definite Integral of Exponential Function

    1. The problem statement, all variables and given/known data
    I have an integral that I need to solve for a quantum physics problem

    [tex]\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx[/tex]

    How would I go about solving this thing?
     
  2. jcsd
  3. Feb 19, 2008 #2
    Split it into two intervals, i.e. [itex](-\infty,0),\,(0,\infty)[/itex] and make a change or variables in the first one [itex]x\to-x[/itex]
     
  4. Feb 19, 2008 #3

    [tex]\int^{\infty}_{-\infty}e^{-a|x| - ikx}dx[/tex]

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

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

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

    Are these steps what you are talking about?
    What would I do from here?
     
  5. Feb 19, 2008 #4
    Since [itex]x\in(0,\infty)\Rightarrow |x|=x[/itex]. Now combine the two integrals.
     
  6. Feb 19, 2008 #5
    Oh...duh....thank you
     
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