Graf's Addition Theorem

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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 in Abramowitz and Stegun 1964 online. We express the theorem in the following form

[math] 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]C_\nu[/math] can represent any of the Bessel functions [math]\,\!J_\nu[/math], [math]\,\!I_\nu[/math], [math]\,\!Y_\nu[/math], [math]\,\!K_\nu[/math], [math]H_\nu^{(1)}[/math], and [math]H_\nu^{(2)}[/math], [math](r_j,\theta_j)\,\![/math] and [math](r_l,\theta_l)\,\![/math] are polar coordinates centred at two different positions with global coordinates [math]\boldsymbol{O}_j [/math], [math] \boldsymbol{O}_l [/math], and [math](R_{jl},\vartheta_{jl})[/math] are the polar coordinates of [math] \boldsymbol{O}_l [/math] with respect to [math] \boldsymbol{O}_j [/math]. This expression is valid only provided that [math]\,\!r_l \lt R_{jl}[/math] ( although this restriction is unnecessary if [math]\,\!C=J[/math] and [math]\,\!\nu[/math] is an integer).

Explicit versions of the theorem are given below,

[math] 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] 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.