Quantum Mechanics Homework Help - Integrals Giving Trouble

[jcain6]

While doing some quantum mechanics homework I came across 3 integrals that are giving me trouble.


1) dx/(a^2+x^2)^2 from negative infinity to infinity

2) (x^2 dx)/(a^2+x^2)^2 from negative infinity to infinity

3) (sinkx)^2/x^2 from 0 to infinity

I would appreciate any help that anyone has to offer. #3 resembles the Dirichlet Integral (if we drop off the squares) and then it would be pi/2. All k's and a's are just constants.

Thanks,

jcain6

 

[amcavoy]

While doing some quantum mechanics homework I came across 3 integrals that are giving me trouble.


1) dx/(a^2+x^2)^2 from negative infinity to infinity

2) (x^2 dx)/(a^2+x^2)^2 from negative infinity to infinity

3) (sinkx)^2/x^2 from 0 to infinity

I would appreciate any help that anyone has to offer. #3 resembles the Dirichlet Integral (if we drop off the squares) and then it would be pi/2. All k's and a's are just constants.

Thanks,

jcain6

For the first one, try $x=a\tan{\theta}$. That will be really messy though, there must be a better way...

 

[lurflurf]

For (2) use "integration by parts"
$$\int_a^b u \ dv=uv|_a^b-\int_a^b v \ du$$
chose u=x dv=x/(x^2+a^2)^2
for (1) note
a^2(1)+(2)=pi/a
for (1) and (2) I assume a>0
for (3)
differentiate w/ respect k to obtain a Dirichlet Integral
note (3)=0 if k=0 then integrate w/respect k to find (3)

 

[Spinny]

I suggest you try to use some complex analysis. It seemed to work for me, on the first one anyway.

 

[amcavoy]

I suggest you try to use some complex analysis. It seemed to work for me, on the first one anyway.

I'm curious. How did you do this?

 

[lurflurf]

I'm curious. How did you do this?

Residue Theorem
$$\oint_C f(z) \ dz=2\pi i\sum res(f(z))_{z=a}$$
In words the Integral of a function around a closed contour equals the sum of the residues of the sigularities of the function times 2pi*i. For (1) and (2) the functions themselves with a contour of a large semicircle in the upper half plan works fine. The sine one is a bit tricker use f(z)=1-exp(2kiz) and a contour of semicircle in upperhalf plane with indentation at origin. There is however no need to use the residue theorem as I above noted the easy way to do them.