- #1
spaghetti3451
- 1,344
- 33
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.
##\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.