Integrating expression with two Abs terms

[Baggio]

Hi,

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

$$\int^{\infty}_{-\infty}{dt.e^{-|t + \tau +a| - |t-a|}$$

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 :)

 

[EnumaElish]

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.

 

[Baggio]

I can't seem to get it... I always end up with just a single point where a sign change occurs

 

[EnumaElish]

Okay; so you have (-infinity, t0] and (t0, +infinity).

 

[Baggio]

Hi,

I tried with the followign example

$$\int^{\infty}_{-\infty}{dt.e^{-|t + 5| - |t|}$$

And I get $$1+5e^{-5}$$

I ran tried evaluating it in maple and it says the answer is $$6e^{-5}$$

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.

 

[Baggio]

I also tried with -inf-> -5/2 and -5/2 -> inf and this just gave me 1

 

[EnumaElish]

With a = 0 and $\tau$ = 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.

 

[Baggio]

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

$$\int^{\infty}_{-\infty}{dt.e^{-|t + \tau +a| - |t-a|}$$

and where tau and 'a' can take any real value -ve or +ve