Integrating expression with two Abs terms

  • Thread starter Baggio
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  • #1
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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 :)
 
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Answers and Replies

  • #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.
 
  • #3
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I can't seem to get it... I always end up with just a single point where a sign change occurs
 
  • #4
EnumaElish
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Okay; so you have (-infinity, t0] and (t0, +infinity).
 
  • #5
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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.
 
  • #6
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I also tried with -inf-> -5/2 and -5/2 -> inf and this just gave me 1
 
  • #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.
 
  • #8
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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
 
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