Jacobi–Anger expansion

Remarks

The generating function of the Bessel functions is $\exp\left( \frac{z}{2} \left( t - \frac{1}{t} \right) \right) = \sum_{n \in \Z} J_n(z)\, t^n$ for $t \ne 0$, obtained by multiplying the series of $e^{zt/2}$ and $e^{-z/(2t)}$. Putting $t = e^{i\theta}$ gives the second formula, since $\frac{1}{2}(t - t^{-1}) = i\sin\theta$. Replacing $\theta$ by $\theta + \frac{\pi}{2}$ turns $\sin\theta$ into $\cos\theta$ and gives the first formula. So the expansion is the Fourier series of the periodic function $\theta \mapsto e^{iz\sin\theta}$, and its Fourier coefficients are $J_n(z)$. Example: at $\theta = 0$ the second formula gives $1 = \sum_{n} J_n(z) = J_0(z) + 2\sum_{k \ge 1} J_{2k}(z)$, using $J_{-n} = (-1)^n J_n$.

Going deeper

For a student

Taking real and imaginary parts gives expansions of $\cos(z\sin\theta)$ and $\sin(z\sin\theta)$ used in frequency modulation, where the strengths of the sidebands are Bessel values. Reading off a Fourier coefficient gives Bessel's integral for $J_n$. In two-dimensional scattering it expands an incident plane wave in cylindrical waves, the first step of the partial-wave method. Multiplying two such expansions gives addition theorems for Bessel functions.