A Integration using delta function and step function

1. Oct 7, 2016

spaghetti3451

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.

2. Oct 8, 2016

andrewkirk

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.

3. Oct 8, 2016

spaghetti3451

Would you recommend me to post this on the Relativity forum?

This question is better suited for that forum.

4. Oct 8, 2016

Fightfish

$$\delta(x^2 - \alpha^2) = \frac{1}{2|\alpha|} \left[\delta(x-\alpha) + \delta(x+\alpha) \right] \neq \frac{1}{2|\alpha|} \left[\delta( (x-\alpha) + (x+\alpha)) \right]$$ 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: Oct 8, 2016