Deriving Bessel Function Equation with Basic Relation

[shaun_chou]

Homework Statement

Known formula:J_0(k\sqrt{\rho^2+\rho'^2-\rho\rho'\cos\phi})=\sum e^{im\phi}J_m(k\rho)J_m(k\rho')
I can't derive to next equation which is e^{ik\rho\cos\phi}=\sum i^me^{im\phi}J_m(k\rho)

Homework Equations

Can anyone help me? Thanks a lot!

The Attempt at a Solution

 

[Pengwuino]

The formula for the first bessel function won't help. I believe this can be proven by looking at the expansion of e^{ik\rho sin(\phi)} in terms of bessel functions.

 

[caduceus]

You can use the basic Bessel relation, i.e;

let k\rho=x

e^{(x/2)(t-1/t)}=\sum J_n(x) t^n

then make the transformation t=e^{i\alpha} st. \alpha=\theta+\pi/2

and then substitute them all in the Bessel relation, then you can obtain the given result.