A Integration using delta function and step function

1,340
30
I would like to evaluate the following integral:

##\displaystyle{\int_{-\infty}^{\infty} dp^{0}\ \delta(p^{2}-m^{2})\ \theta(p^{0})}##

##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0})^{2}-\omega^{2}]\ \theta(p^{0})}##

##\displaystyle{= \int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0}+\omega)(p^{0}-\omega)]\ \theta(p^{0})}##

##\displaystyle{= \frac{1}{2\omega}\int_{-\infty}^{\infty} dp^{0}\ \delta[(p^{0}+\omega)+(p^{0}-\omega)]\ \theta(p^{0})}##

##\displaystyle{= \frac{1}{2\omega}\int_{-\infty}^{\infty} dp^{0}\ \delta(2p^{0})\theta(p^{0})}##

Have I made a mistake somewhere?

The answer is supposed to be ##\frac{1}{\omega}## but I'm not sure how to proceed next.
 

andrewkirk

Science Advisor
Homework Helper
Insights Author
Gold Member
3,770
1,384
I don't follow some of your steps so far. But if you want to be able to get rid of the integral, try a change of variable, making the variable over which we are integrating ##u=2p^0##.

The integral can then be evaluated quite simply. However I then get the answer ##1/4\omega##, which is why I am wondering about your earlier steps.
 
1,340
30
Would you recommend me to post this on the Relativity forum?

This question is better suited for that forum.
 
954
117
[tex]
\delta(x^2 - \alpha^2) = \frac{1}{2|\alpha|} \left[\delta(x-\alpha) + \delta(x+\alpha) \right] \neq \frac{1}{2|\alpha|} \left[\delta( (x-\alpha) + (x+\alpha)) \right]
[/tex] which is what you seem to have done.
Are you sure the answer is supposed to be ##1/\omega## and not ##1/2\omega## though?
 
Last edited:

Related Threads for: Integration using delta function and step function

  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
20
Views
2K
  • Last Post
Replies
14
Views
6K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
7
Views
7K

Hot Threads

Top