This is the statement I read in a paper. By $\delta$, I mean the Dirac delta function and by $\theta$ (or $H$) I mean the Heaviside step function.
I do not know the reason behind, but I would without hesitation write
$$ \int_{-\infty}^{\infty } \delta(x) H(x) dx = 1/2 ,$$
and
$$ \int_{-\infty}^{\infty } \delta(x) H(x) f(x) dx = f(0)/2 $$
for any smooth function $f$. This worked for several problems I encountered in quantum mechanics. They lead to right answers. That is why I am shocked by the statement above.
Could anyone point out the error for me?
Why $\frac12$?
Using $\delta=H'$ and $H^n = H$ we get $$ \int_{-\infty}^{\infty} \delta(x) \, H(x) \, dx = \int_{-\infty}^{\infty} H'(x) \, H(x)^n \, dx = \int_{-\infty}^{\infty} \frac{1}{n+1} (H(x)^{n+1})' \, dx \\ = \frac{1}{n+1} \left[H(x)^{n+1}\right]_{-\infty}^{\infty} = \frac{1}{n+1}. $$ Only for $n=1$ we get $\frac12$. Where's the error?