Difference between revisions of "Graf's Addition Theorem"

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Graf's addition theorem for Bessel functions, given in
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Graf's addition theorem for Bessel functions is given in
[[Abramowitz and Stegun 1964]], is
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[[Abramowitz and Stegun 1964]]. It is a special case of a general addition theorem called Neumann's addition theorem. Details
 +
can be found in [http://www.math.sfu.ca/~cbm/aands/page_363.htm Abramowitz and Stegun 1964 online]. We express the theorem
 +
in the following form
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<center><math>
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C_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} =
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\sum_{\mu = - \infty}^{\infty} C_{\nu + \mu} (\eta R_{jl}) \,
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J_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})},
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\quad j \neq l,
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</math></center>
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where <math>C_\nu</math> can represent any of the [http://en.wikipedia.org/wiki/Bessel_function Bessel functions]
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<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>,
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<math>(r_j,\theta_j)\,\!</math> and <math>(r_l,\theta_l)\,\!</math> are polar coordinates centred at two different positions
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with global coordinates <math>\boldsymbol{O}_j </math>, <math> \boldsymbol{O}_l </math>, and
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<math>(R_{jl},\vartheta_{jl})</math> are the polar coordinates of <math> \boldsymbol{O}_l </math> with respect to <math> \boldsymbol{O}_j </math>.
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This expression is valid only provided that <math>\,\!r_l < R_{jl}</math> (
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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,
 
<center><math>  
 
<center><math>  
 
H_\nu^{(1)}(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} =
 
H_\nu^{(1)}(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} =
Line 12: Line 29:
 
(\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,
 
(\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,
 
</math></center>
 
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>.
+
This theorem form the basis for [[Kagemoto and Yue Interaction Theory]].
Here, <math>(R_{jl},\varphi_{jl})</math>  are the polar coordinates of the mean centre position of <math>\Delta_{l}</math> in the local coordinates of <math>\Delta_{j}</math>.
 
 
 
This theorem form the basis for [[Kagemoto and Yue Interaction Theory]]
 
  
 
[[Category:Numerical Methods]]
 
[[Category:Numerical Methods]]
[[Category:Linear Water-Wave Theory]]
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[[Category:Interaction Theory]]

Latest revision as of 05:42, 28 April 2009

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]\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 with global coordinates [math]\displaystyle{ \boldsymbol{O}_j }[/math], [math]\displaystyle{ \boldsymbol{O}_l }[/math], and [math]\displaystyle{ (R_{jl},\vartheta_{jl}) }[/math] are the polar coordinates of [math]\displaystyle{ \boldsymbol{O}_l }[/math] with respect to [math]\displaystyle{ \boldsymbol{O}_j }[/math]. This expression is valid only provided that [math]\displaystyle{ \,\!r_l \lt R_{jl} }[/math] ( although this restriction is unnecessary 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.