Recast of an expression containing Diracs [tex]\delta[/tex]-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 ?

 

[HallsofIvy]

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.

 

[johnpatitucci]

Thanks for your reply.

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?