Graf's Addition Theorem

Graf's addition theorem for Bessel functions is given in Abramowitz and Stegun 1964. It is a special case of a general addition theorem called Neumann's addition theorem. Details can be found Abramowitz and Stegun 1964 online. We express the theorem in the following form

[math]\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, }[/math]

where [math]\displaystyle{ C_\nu }[/math] can represent any of the Bessel functions [math]\displaystyle{ J_\nu }[/math], [math]\displaystyle{ I_\nu }[/math], [math]\displaystyle{ Y_\nu }[/math], [math]\displaystyle{ K_\nu }[/math], [math]\displaystyle{ H_\nu^{(1)} }[/math], and [math]\displaystyle{ H_\nu^{(2)} }[/math], [math]\displaystyle{ (r_j,\theta_j) }[/math] and [math]\displaystyle{ (r_l,\theta_l) }[/math] are polar coordinates centred at two different positions\ and [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] is the polar coordinates of the centre position of the polar coordinate system [math]\displaystyle{ l }[/math] in the coordinates of the polar coordinate system [math]\displaystyle{ j }[/math]. This expression is valid only provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math] ( with the exception that this restriction is not necessary if [math]\displaystyle{ C=J }[/math] and [math]\displaystyle{ \nu }[/math] is an integer).

Explicit versions of the theorem are given below,

[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]

This theorem form the basis for Kagemoto and Yue Interaction Theory.