Difference between revisions of "Graf's Addition Theorem"

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which is valid provided that <math>r_l < R_{jl}</math>.  
 
which is valid provided that <math>r_l < R_{jl}</math>.  
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This theorem form the basis for [[Kagemoto and Yue Interaction Theory]]
  
 
[[Category:Numerical Methods]]
 
[[Category:Numerical Methods]]
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Revision as of 23:17, 7 June 2006

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].

This theorem form the basis for Kagemoto and Yue Interaction Theory