# Recast of an expression containing Diracs $$\delta$$-function

johnpatitucci
Hey there,

I got a problem with one recast of an expression which pops up by considering the completeness relation of common spherical harmonics:

$$\sum_{l=0}^{\infty} \sum_{m=-l}^{+l} Y_{lm}(\theta,\phi)Y^{*}_{lm}(\theta^{\prime},\phi^{\prime}) = \frac{1}{sin(\theta)} \delta(\theta - \theta^{\prime}) \delta(\phi - \phi^{\prime})$$

The question is about the rhs which is sometimes recast like

$$\frac{1}{sin(\theta)} \delta(\theta - \theta^{\prime}) \delta(\phi - \phi^{\prime}) = \delta(\cos(\theta) - \cos(\theta^{\prime})) \delta(\phi - \phi^{\prime})$$

After several attempts I just can't explain how to justify that rearrangement. Could you help me please ?

It isn't quite the same thing, unless you restrict $\phi$ to be between $-\pi/2$ and $\pi/2$. In that range, cosine is "one to one" so that $cos(\phi)- cos(\phi)$ if and only if $\phi- \phi'$ which is all that matters for the delta function.
I still don't get it. Cos(x) is not injective between -$$\pi /2$$ and $$\pi /2$$ as it is symmetric within this range (take e.g. $\pm \frac{\pi}{4}$ and you obtain $\cos(\pm\frac{\pi}{4}) = \frac{1}{\sqrt(2)}$ ). It would be true if $$0 \leq x \leq \pi$$. But nevertheless, what happens to the $$1 / \sin(\theta)$$ term?