Spherical bessel functions addition theorems

[Knockout]

I really need to prove eq. 10.1.45 and 10.1.46 of Abramowitz and Stegun Handbook on Mathematical functions. Is an expansion of e^(aR)/R in terms of Special Functions! Any help will be appreciated.

 

[Knockout]

Ok I'll write it down. I need to prove:
\frac{e^{-ikR}}{R}=\sum [(2n+1)j_{n}(kb)h_{n}(kr)P_{n}(cos \theta)]
where:
R=\sqrt{r^2+b^2-2brcos(\theta)} and the sum goes from zero to infinity over n.
I know it's a particular case of gegenbauer addition theorem. I understand what it means. I only need a simple proof. (I've seen a proof using the green function of the Helmholtz equation, but I'm sure it's even simpler than that.)

Oh j are the first spherical bessel functions and h are the second Hankel spherical functions.
Any help would be appreciated.

 

[Ben Niehoff]

It might help to use

j_n(r) = (-1)^n r^n \left( \frac{d}{r \, dr} \right)^n \frac{\sin r}{r}

I'm not sure. There is a similar relation for the spherical Hankel functions.

 

[denny137]

Knockout,
I'm interested in finding a proof too for that relation too.
Did you find a simple proof of it? Also, do you know
of a reference that does a green's function proof through the Helmholtz eqn?
Thanks

 

[Knockout]

Denny sorry for my late answer. I just came back from a long holiday. The green's function proof was in several books. One I can remember was Fundamentals of Mathematical Physics by Edgar Kraut. My problem with those proofs is that they propose a magical expansion from nowhere (which is ok), but i was looking for something more didactical. You can find a more powerful and difficult version of the proof as the Gegenbauer addition theorem. This you can check it in the famous book: A treatise on the Theory of Bessel Functions by George Neville Watson. However ,I haven't suceeded in finding a simple didactical proof, and I'm still looking for one. If you already found one please let me know. I'm working on some class notes, and I'm trying to make them as simple as possible.

 

[denny137]

Thanks for the reply. Yes, I agree. I think the proof of
the Gegenbauer addition theorem in Watson took too long to go
through. Though, I did find a simple Green's function proof of the Helmoltz eqn. that
I thought was not too 'magical' in:
Mathematics of Classical and Quantum Physics (Paperback)
by Frederick W. Byron (Author), Robert W. Fuller

In there, they show e(ik*R)/R is a Green's function to the Helmoltz eqn.
in free space through a Fourier transform method. Then, they also show
the sum form of the Green's function involving spherical bessel function can
also be obtained by separation of variables after satisfying boundary conditions.
I thought those 2 derivations were relatively straightforward and I was ok with
simply equating the 2 expressions. (I think this is the idea, but its been a month
or two since I looked at it.)


Denny

 

[Knockout]

Thanks Denny. Yesterday, I solve it in a simple way for my notes. I also saw Byron's book, and it's a nice way to solve it. Thanks again anyway, hope this thread helps some other lost soul.