Undergrad What happens when we replace the Dirac Delta function with a sine function?

[Kyle Nemeth]

If we were to replace δ(x), the orginal Dirac Delta, with δ(sin(ωx)), what would be the result?

Would we have an infinite spike everywhere on the graph of sinx where x is a multiple integer of π/ω? and 0 everywhere else?

I apologize in advance if I had posted in the wrong category.

 

[Mark44]

If we were to replace δ(x), the orginal Dirac Delta, with δ(sin(ωx)), what would be the result?

Would we have an infinite spike everywhere on the graph of sinx where x is a multiple integer of π/ω? and 0 everywhere else?

Seems reasonable that $\delta(\sin(\omega x))$ would be infinite wherever $\omega x$ is an odd multiple of $\pi/2$, and zero everywhere else. In other words, where $x = \frac{(2k + 1)\pi}{2\omega}$, with k in the integers.

 

[Orodruin]

Yes. For a fuction $f(x)$ with a countable set of zeroes $x_i$, it holds that
$$
\delta(f(x)) = \sum_i \frac{\delta(x-x_i)}{|f’(x_i)|}
$$

 

[Orodruin]

Seems reasonable that $\delta(\sin(\omega x))$ would be infinite wherever $\omega x$ is an odd multiple of $\pi/2$, and zero everywhere else. In other words, where $x = \frac{(2k + 1)\pi}{2\omega}$, with k in the integers.

The zeroes of the sine function are integer multiples of pi.

 

[Kyle Nemeth]

Thank you guys for answering my question. Very much appreciated.

 

[andrewkirk]

Speaking very loosely, the answer is yes.

Speaking more carefully, the dirac delta is a distribution, not a function, and only gives a value when included in an integral. That is
$\int_a^b\delta(x)f(x)dx$ is equal to $f(0)$ if $a\leq 0 \leq b$ and to zero otherwise. So your question could be expressed more precisely as:
what is the value of $\int_a^b \delta(\sin\omega x)f(x)dx$?

This can be answered by substitution. Set $u=\sin\omega x$ so that $du=\omega\cos x\,dx=\omega\sqrt{1-u^2}\,dx$. Then the Dirac delta part of the integrand becomes just $\delta(u)$.

We'll end up with an answer that is equal to the sum of the values of $f(x)$ for all values of $x$ in $[a,b]$ that are multiples of $\pi/\omega$.

 

[Mark44]

The zeroes of the sine function are integer multiples of pi.

Doh!