# Homework Help: Bessel Function simplification

1. Feb 25, 2010

### oliphant

1. The problem statement, all variables and given/known data

I'm trying to convert $$s(t) = sin(2 \pi f_c t + I sin[2 \pi f_1 t + I_2 sin\{2 \pi f_2 t\}])$$ into Bessel functions of the form

$$s(t) = \sum_k J_k(I_1) \times J_n (k I_2) sin(2\pi [f_c + k_1 f_1 + n f_2]t)$$

2. Relevant equations

Standard trigonometric equation for sin addition sin(x+y).

also, $$cos(z\ sint) &=& J_0(z) + J_2(z)cos(2t) + J_4(z)cos(4t) + \ldots \\ &=& J_0(z) + \sum_{n=1}^\infty J_{2n}(z)cos(2nt) \label{as:3}$$

and

$$sin(z\ sint) &=& J_1 (z) sin(t) + J_3(z) sin(3t) + J_5(z) sin(5t) + \ldots \\ &=& \sum_{n=0}^\infty J_{2n+1}(z)sin((2n+1)t) \label{as:2}$$

3. The attempt at a solution

I can only get far as expanding the equation for sin(x+y), but then it gets a bit confusing when I have to do sin of a sum of infinite Bessel functions. Can anyone offer a hand?