# Graf's Addition Theorem

$\displaystyle{ C_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} C_{\nu + \mu} (\eta R_{jl}) \, J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l, }$
where $\displaystyle{ C_\nu }$ can represent any of the Bessel functions $\displaystyle{ \,\!J_\nu }$, $\displaystyle{ \,\!I_\nu }$, $\displaystyle{ \,\!Y_\nu }$, $\displaystyle{ \,\!K_\nu }$, $\displaystyle{ H_\nu^{(1)} }$, and $\displaystyle{ H_\nu^{(2)} }$, $\displaystyle{ (r_j,\theta_j)\,\! }$ and $\displaystyle{ (r_l,\theta_l)\,\! }$ are polar coordinates centred at two different positions with global coordinates $\displaystyle{ \boldsymbol{O}_j }$, $\displaystyle{ \boldsymbol{O}_l }$, and $\displaystyle{ (R_{jl},\vartheta_{jl}) }$ are the polar coordinates of $\displaystyle{ \boldsymbol{O}_l }$ with respect to $\displaystyle{ \boldsymbol{O}_j }$. This expression is valid only provided that $\displaystyle{ \,\!r_l \lt R_{jl} }$ ( although this restriction is unnecessary if $\displaystyle{ \,\!C=J }$ and $\displaystyle{ \,\!\nu }$ is an integer).
$\displaystyle{ H_\nu^{(1)}(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\eta R_{jl}) \, J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l, }$
$\displaystyle{ K_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} = \sum_{\mu = - \infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l, }$