# Bessel function equivalence

1. Dec 18, 2009

### shaun_chou

1. The problem statement, all variables and given/known data
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 $$e^{ik\rho\cos\phi}=\sum i^me^{im\phi}J_m(k\rho)$$

2. Relevant equations
Can anyone help me? Thanks a lot!!

3. The attempt at a solution

2. Dec 18, 2009

### 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.

3. Dec 30, 2009

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.