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Integrating expression with two Abs terms

  1. Dec 1, 2007 #1
    Hi,

    I'm trying to integrate an expression I derived from multiplying the Fourier transform of two Lorentzians together. The expression is

    [tex]\int^{\infty}_{-\infty}{dt.e^{-|t + \tau +a| - |t-a|}[/tex]

    How do you go about solving this? I tried splitting it up but you have two different values of t where a sign change occurs...

    Any help would be appreciated :)
     
    Last edited: Dec 1, 2007
  2. jcsd
  3. Dec 1, 2007 #2

    EnumaElish

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    Identify all points at which the exponent changes sign. Suppose you found 4 points t1 - t4. (I made it up). Then integrate over (-infinity, t1], (t1, t2], (t2, t3], (t3, +infinity) separately.
     
  4. Dec 1, 2007 #3
    I can't seem to get it... I always end up with just a single point where a sign change occurs
     
  5. Dec 1, 2007 #4

    EnumaElish

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    Okay; so you have (-infinity, t0] and (t0, +infinity).
     
  6. Dec 2, 2007 #5
    Hi,

    I tried with the followign example

    [tex]\int^{\infty}_{-\infty}{dt.e^{-|t + 5| - |t|}[/tex]

    And I get [tex]1+5e^{-5}[/tex]

    I ran tried evaluating it in maple and it says the answer is [tex]6e^{-5}[/tex]

    I don't see what I'm doing wrong I integrated from -inf to -5/2 then -5/2 to +5/2 then 5/2 to inf.
     
  7. Dec 2, 2007 #6
    I also tried with -inf-> -5/2 and -5/2 -> inf and this just gave me 1
     
  8. Dec 2, 2007 #7

    EnumaElish

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    With a = 0 and [itex]\tau[/itex] = 5, the plot of -|t + 5| - |t| is increasing for t < -5, constant at -5 for -5 < t < 0, then decreasing for t > 0. So you have 3 regions.
     
  9. Dec 2, 2007 #8
    Yep thanks for that managed to work it out... It's easy to see the limits when you plot it out.

    Though what if you have something like

    [tex]\int^{\infty}_{-\infty}{dt.e^{-|t + \tau +a| - |t-a|}[/tex]

    and where tau and 'a' can take any real value -ve or +ve
     
    Last edited: Dec 2, 2007
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