Graf's addition theorem for Bessel functions, given in
Abramowitz and Stegun 1964, is
[math]\displaystyle{
H_\nu^{(1)}(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} =
\sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\eta R_{jl}) \,
J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})},
\quad j \neq l,
}[/math]
[math]\displaystyle{
K_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = -
\infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu
(\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,
}[/math]
which is valid provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math].
