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Difference between revisions of "Toroidal harmonics"

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Functions of the points on a torus that arise when solving the [[Laplace equation|Laplace equation]] by the method of separation of variables (cf. [[Separation of variables, method of|Separation of variables, method of]]) in [[Toroidal coordinates|toroidal coordinates]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932801.png" />. A [[Harmonic function|harmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932802.png" />, which is a solution of the Laplace equation, can be written as a series
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932804.png" /></td> </tr></table>
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Functions of the points on a torus that arise when solving the [[Laplace equation|Laplace equation]] by the method of separation of variables (cf. [[Separation of variables, method of|Separation of variables, method of]]) in [[Toroidal coordinates|toroidal coordinates]]  $  ( \sigma , \tau , \phi ) $.  
 +
A [[Harmonic function|harmonic function]]  $  h = h ( \sigma , \tau , \phi ) $,
 +
which is a solution of the Laplace equation, can be written as a series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932805.png" /></td> </tr></table>
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$$ \tag{* }
 +
= \sqrt {\cosh  \tau - \cos  \sigma } \times
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932807.png" /> are the associated [[Legendre functions|Legendre functions]] with half-integer index. By setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932808.png" /> one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t0932809.png" />, is sometimes called a spatial toroidal harmonic.
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$$
 +
\times
 +
\sum _ {j, k = 0 } ^  \infty  [ A _ {jk} P _ {j - 1/2 }  ^ {(} k) ( \cosh  \tau ) + B _ {jk} Q _ {j - 1/2 }  ^ {(} k) ( \cosh  \tau )] \times
 +
$$
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 +
$$
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\times
 +
( a _ {k}  \cos  k \sigma + b _ {k}  \sin  k \sigma )
 +
( c _ {j}  \cos  j \phi + d _ {j}  \sin  j \phi ),
 +
$$
 +
 
 +
where the  $  P _ {j - 1/2 }  ^ {(} k) $,  
 +
$  Q _ {j - 1/2 }  ^ {(} k) $
 +
are the associated [[Legendre functions|Legendre functions]] with half-integer index. By setting $  \tau = \tau _ {0} $
 +
one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables $  ( \sigma , \tau , \phi ) $,  
 +
is sometimes called a spatial toroidal harmonic.
  
 
The series (*) is used in the solution of boundary value problems in toroidal coordinates, taking into account the expansion
 
The series (*) is used in the solution of boundary value problems in toroidal coordinates, taking into account the expansion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328010.png" /></td> </tr></table>
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$$
 +
{
 +
\frac{1}{\sqrt {\cosh  \tau - \cos  \sigma } }
 +
} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328011.png" /></td> </tr></table>
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$$
 +
= \
 +
{
 +
\frac{\sqrt 2 } \pi
 +
} \left ( Q _ {- 1/2 }  (
 +
\cosh  \tau ) + 2 \sum _ {k = 1 } ^  \infty  Q _ {k
 +
- 1/2 }  ( \cosh  \tau )  \cos  k \sigma \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093280/t09328012.png" /> is the Legendre function of the second kind.
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where $  Q _ {k - 1/2 }  $
 +
is the Legendre function of the second kind.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Morse,  H. Feshbach,  "Methods of theoretical physics" , '''1–2''' , McGraw-Hill  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tichonoff,  A.A. Samarskii,  "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Morse,  H. Feshbach,  "Methods of theoretical physics" , '''1–2''' , McGraw-Hill  (1953)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)  (Formula 3.10 (3))</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)  (Formula 3.10 (3))</TD></TR></table>

Revision as of 08:26, 6 June 2020


Functions of the points on a torus that arise when solving the Laplace equation by the method of separation of variables (cf. Separation of variables, method of) in toroidal coordinates $ ( \sigma , \tau , \phi ) $. A harmonic function $ h = h ( \sigma , \tau , \phi ) $, which is a solution of the Laplace equation, can be written as a series

$$ \tag{* } h = \sqrt {\cosh \tau - \cos \sigma } \times $$

$$ \times \sum _ {j, k = 0 } ^ \infty [ A _ {jk} P _ {j - 1/2 } ^ {(} k) ( \cosh \tau ) + B _ {jk} Q _ {j - 1/2 } ^ {(} k) ( \cosh \tau )] \times $$

$$ \times ( a _ {k} \cos k \sigma + b _ {k} \sin k \sigma ) ( c _ {j} \cos j \phi + d _ {j} \sin j \phi ), $$

where the $ P _ {j - 1/2 } ^ {(} k) $, $ Q _ {j - 1/2 } ^ {(} k) $ are the associated Legendre functions with half-integer index. By setting $ \tau = \tau _ {0} $ one obtains a toroidal harmonic or a surface toroidal harmonic, this in contrast with the expression (*) which, as a function of the three variables $ ( \sigma , \tau , \phi ) $, is sometimes called a spatial toroidal harmonic.

The series (*) is used in the solution of boundary value problems in toroidal coordinates, taking into account the expansion

$$ { \frac{1}{\sqrt {\cosh \tau - \cos \sigma } } } = $$

$$ = \ { \frac{\sqrt 2 } \pi } \left ( Q _ {- 1/2 } ( \cosh \tau ) + 2 \sum _ {k = 1 } ^ \infty Q _ {k - 1/2 } ( \cosh \tau ) \cos k \sigma \right ) , $$

where $ Q _ {k - 1/2 } $ is the Legendre function of the second kind.

References

[1] A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[2] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)

Comments

References

[a1] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953) (Formula 3.10 (3))
How to Cite This Entry:
Toroidal harmonics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toroidal_harmonics&oldid=48997
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article