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

  • #1
Hey there,


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

[tex]\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})[/tex]

The question is about the rhs which is sometimes recast like

[tex]\frac{1}{sin(\theta)} \delta(\theta - \theta^{\prime}) \delta(\phi - \phi^{\prime}) = \delta(\cos(\theta) - \cos(\theta^{\prime})) \delta(\phi - \phi^{\prime})[/tex]

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

Answers and Replies

  • #2
HallsofIvy
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It isn't quite the same thing, unless you restrict [itex]\phi[/itex] to be between [itex]-\pi/2[/itex] and [itex]\pi/2[/itex]. In that range, cosine is "one to one" so that [itex]cos(\phi)- cos(\phi)[/itex] if and only if [itex]\phi- \phi'[/itex] which is all that matters for the delta function.
 
  • #3
Thanks for your reply.

I still don't get it. Cos(x) is not injective between -[tex]\pi /2[/tex] and [tex]\pi /2[/tex] as it is symmetric within this range (take e.g. [itex] \pm \frac{\pi}{4}[/itex] and you obtain [itex] \cos(\pm\frac{\pi}{4}) = \frac{1}{\sqrt(2)} [/itex] ). It would be true if [tex] 0 \leq x \leq \pi [/tex]. But nevertheless, what happens to the [tex] 1 / \sin(\theta)[/tex] term?
 

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