# Dirac delta and trig functions

I ran across the identity: $$\delta(f(x)) = \sum_i \frac{\delta(x - x_i)}{|f^\prime(x_i)|}$$ where $$\delta(x)$$ is the Dirac delta and $$x_i$$ are the roots of some function $$f(x)$$. Now, I have seen in physics textbooks people use $$\delta(\vec{r} - \vec{r}_0) = \frac{1}{r^2\sin\theta}\delta(r-r_0)\delta(\theta - \theta_0)\delta(\phi - \phi_0)$$ as the Dirac delta in spherical coordinates. Where I am deeply confused is that some replace the $$\theta$$ part with $$\delta(\theta - \theta_0)/\sin\theta \rightarrow \delta(\cos\theta - \cos\theta_0)$$. I thought I was comfortable with this transformation because of the above rule, but I not so sure anymore. Take for example, a line defined as $$\delta(\cos^2\theta - 1)$$. By the above identity, I should be able to transform this to $$\delta(\cos^2\theta - 1) = \frac{\delta(\theta)}{|2\sin(0)|} + \frac{\delta(\theta - \pi)}{|2\sin\pi|},$$ but as you can see, this function blows up at the poles. So this means we cannot really write $$\frac{\delta(\theta - \theta_0)}{\sin\theta} = \delta(\cos\theta - \cos\theta_0),$$ correct? Is this one of those common physics habits of fudging up the math? I see the $$\delta(\cos\theta)$$ so much, I just feel that I am grossly misunderstanding something, but it is not easy to Google. Any insight is greatly appreciated! Thanks.

• Which physics textbooks? Which pages? Commented Apr 7, 2023 at 19:44
• @Qmechanic, I remember seeing it in undergraduate E&M, but I cannot readily find it. I know it is in Barton's "Elements of Green’s Functions and Propagation". Also, Eq. (19) in this this paper and Eq. (13) in this web book Commented Apr 7, 2023 at 21:57
• In your original setting you need to avoid $\theta_0$ being a multiple of $\pi$ I guess. Which makes sense anyway because this causes a division by 0 either way.
– Ian
Commented Apr 8, 2023 at 3:51

The definition of the composition of the Dirac Delta, $$\delta \circ f$$, with a function $$f$$ assumes that $$(1)$$ $$f$$ is continuously differentiable and $$(2)$$ $$f'$$ is nowhere zero. Under such assumptions then if $$f(x_i)=0$$ for $$i=1, \cdots N$$ we define $$\delta\circ f$$ by

$$\delta \circ f=\sum_{i=1}^N \frac{\delta_{x_i}}{|f'(x_i)|}$$

where $$\langle \delta_{x_i}, \phi \rangle =\phi(x_i)$$ for any $$\phi\in C_C^\infty$$.

Then, we have

$$\langle \delta\circ f, \phi\rangle =\sum_{i=1}^N \frac{\phi(x_i)}{| f'(x_i)|}$$

Now, suppose $$f(x)=\cos^2(x)-1$$. Note that $$f(0)=f(\pi)=0$$. Note also $$f'(x)=\sin(2x)$$ so that $$f'(0)=f'(\pi)=0$$. Hence, $$f(x)=\cos^2(x)-1$$ fails to satisfy the assumptions of the definition.

• Thank you very much for this answer. Would you be able to explain to me why so many substitute $\delta(\theta - \theta_0) / \sin\theta$ with $\delta(\cos\theta - \cos\theta_0)$? If you look at my comments above, I cite a couple references where this is used. Commented Apr 8, 2023 at 14:26
• You're welcome. My pleasure. Let $f(\theta)=\cos(\theta)-\cos(\theta_0)$. Then, $f(\theta_0)=0$ and $f'(\theta)=\sin(\theta)$. Now use the definition for the composite Dirac Delta. Commented Apr 8, 2023 at 15:31
• Thanks! I think you are nearing the end of my confusion. One last question, why do you use $f^\prime(\theta)$ instead of $f^\prime(\theta_0)$ if $\theta_0$ is the zero of $f(\theta)$ and the composite Dirac delta is defined only at the zeros? Commented Apr 8, 2023 at 16:24
• I think this is the main source of my confusion. Basically, $\delta(\cos\theta - \cos\theta_0) = \delta(\theta - \theta_0)/|-\sin\theta_0|$ based on the definition of composite Dirac delta, no? So where did the non-constant $\sin\theta$ go? In my mind, $\delta(\theta - \theta_0)/\sin\theta = \delta(\cos\theta - \cos\theta_0)\sin\theta_0/\sin\theta$ based on the definitions provided. Commented Apr 8, 2023 at 16:37
• Note that $f(x)\delta(x-x_0)=f(x_0)\delta(x-x_0)$ Commented Apr 9, 2023 at 15:57