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''ellipsoidal harmonic function''
 
''ellipsoidal harmonic function''
  
 
A function of special form satisfying the [[Lamé equation|Lamé equation]]. If the Lamé equation in algebraic form,
 
A function of special form satisfying the [[Lamé equation|Lamé equation]]. If the Lamé equation in algebraic form,
  
<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/l/l057/l057410/l0574101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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 +
\frac{d  ^ {2} w }{d \xi  ^ {2} }
 +
+
 +
 
 +
\frac{1}{2}
 +
 
 +
\left (
 +
 
 +
\frac{1}{\xi - e _ {1} }
 +
+
 +
 
 +
\frac{1}{\xi - e _ {2} }
 +
+
 +
 
 +
\frac{1}{\xi - e _ {3} }
 +
 
 +
\right )
 +
 
 +
\frac{dw}{d \xi }
 +
=
 +
$$
  
<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/l/l057/l057410/l0574102.png" /></td> </tr></table>
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$$
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= \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l0574103.png" /> is natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l0574104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l0574105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l0574106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l0574107.png" /> are constants, has a solution of one of the following forms:
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\frac{A + n ( n + 1 ) \xi }{4 ( \xi - e _ {1} ) ( \xi - e _ {2} ) ( \xi - e _ {3} ) }
 +
w ,
 +
$$
  
<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/l/l057/l057410/l0574108.png" /></td> </tr></table>
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where  $  n $
 +
is natural number and  $  e _ {1} $,
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$  e _ {2} $,
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$  e _ {3} $,
 +
and  $  A $
 +
are constants, has a solution of one of the following forms:
  
<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/l/l057/l057410/l0574109.png" /></td> </tr></table>
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$$
 +
P ( \xi ) ,
 +
$$
  
<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/l/l057/l057410/l05741010.png" /></td> </tr></table>
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$$
 +
\sqrt {\xi - e _ {i} } P ( \xi ) ,\  i = 1 , 2 , 3 ,
 +
$$
  
<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/l/l057/l057410/l05741011.png" /></td> </tr></table>
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$$
 +
\sqrt {\xi - e _ {i} } \sqrt {\xi - e _ {j} } P (
 +
\xi ) ,\  i , j = 1 , 2 , 3 ,\  i \neq j ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741012.png" /> is a polynomial with leading coefficient one, then this solution is called a Lamé function of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741014.png" /> of the first kind and the first, second, third, or fourth form, respectively.
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$$
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\sqrt {\xi - e _ {1} } \sqrt {\xi - e _ {2} } \sqrt {\xi - e _ {3} } P ( \xi ) ,
 +
$$
  
For fixed even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741018.png" /> there are always values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741019.png" /> (eigen values) such that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741020.png" /> Lamé functions of the first form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741021.png" /> of the third form, with polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741022.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741024.png" />, respectively. For fixed odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741025.png" /> there are always values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741026.png" /> such that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741027.png" /> Lamé functions of the second form and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741028.png" /> of the fourth form, with polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741029.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741031.png" />, respectively. For a given natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741032.png" /> there are altogether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057410/l05741033.png" /> linearly independent Lamé functions.
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where  $  P ( \xi ) $
 +
is a polynomial with leading coefficient one, then this solution is called a Lamé function of degree  $  n $
 +
of the first kind and the first, second, third, or fourth form, respectively.
 +
 
 +
For fixed even $  n $
 +
there are always values of $  A $(
 +
eigen values) such that there are $  ( n + 2 ) / 2 $
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Lamé functions of the first form and $  3 n / 2 $
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of the third form, with polynomials $  P ( \xi ) $
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of degree $  n / 2 $
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and $  ( n - 2 ) / 2 $,  
 +
respectively. For fixed odd $  n $
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there are always values of $  A $
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such that there are $  3 ( n + 1 ) / 2 $
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Lamé functions of the second form and $  ( n- 1)/2 $
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of the fourth form, with polynomials $  P( \xi ) $
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of degree $  ( n- 1)/2 $
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and $  ( n - 3 ) / 2 $,  
 +
respectively. For a given natural number $  n $
 +
there are altogether $  2 n + 1 $
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linearly independent Lamé functions.
  
 
Solutions of equation (*) that are linearly independent with the Lamé functions of the first kind and are obtained by means of the [[Liouville–Ostrogradski formula|Liouville–Ostrogradski formula]] are called Lamé functions of the second kind.
 
Solutions of equation (*) that are linearly independent with the Lamé functions of the first kind and are obtained by means of the [[Liouville–Ostrogradski formula|Liouville–Ostrogradski formula]] are called Lamé functions of the second kind.
  
 
For references see [[Lamé equation|Lamé equation]].
 
For references see [[Lamé equation|Lamé equation]].
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[[Category:Special functions]]

Latest revision as of 22:15, 5 June 2020


ellipsoidal harmonic function

A function of special form satisfying the Lamé equation. If the Lamé equation in algebraic form,

$$ \tag{* } \frac{d ^ {2} w }{d \xi ^ {2} } + \frac{1}{2} \left ( \frac{1}{\xi - e _ {1} } + \frac{1}{\xi - e _ {2} } + \frac{1}{\xi - e _ {3} } \right ) \frac{dw}{d \xi } = $$

$$ = \ \frac{A + n ( n + 1 ) \xi }{4 ( \xi - e _ {1} ) ( \xi - e _ {2} ) ( \xi - e _ {3} ) } w , $$

where $ n $ is natural number and $ e _ {1} $, $ e _ {2} $, $ e _ {3} $, and $ A $ are constants, has a solution of one of the following forms:

$$ P ( \xi ) , $$

$$ \sqrt {\xi - e _ {i} } P ( \xi ) ,\ i = 1 , 2 , 3 , $$

$$ \sqrt {\xi - e _ {i} } \sqrt {\xi - e _ {j} } P ( \xi ) ,\ i , j = 1 , 2 , 3 ,\ i \neq j , $$

$$ \sqrt {\xi - e _ {1} } \sqrt {\xi - e _ {2} } \sqrt {\xi - e _ {3} } P ( \xi ) , $$

where $ P ( \xi ) $ is a polynomial with leading coefficient one, then this solution is called a Lamé function of degree $ n $ of the first kind and the first, second, third, or fourth form, respectively.

For fixed even $ n $ there are always values of $ A $( eigen values) such that there are $ ( n + 2 ) / 2 $ Lamé functions of the first form and $ 3 n / 2 $ of the third form, with polynomials $ P ( \xi ) $ of degree $ n / 2 $ and $ ( n - 2 ) / 2 $, respectively. For fixed odd $ n $ there are always values of $ A $ such that there are $ 3 ( n + 1 ) / 2 $ Lamé functions of the second form and $ ( n- 1)/2 $ of the fourth form, with polynomials $ P( \xi ) $ of degree $ ( n- 1)/2 $ and $ ( n - 3 ) / 2 $, respectively. For a given natural number $ n $ there are altogether $ 2 n + 1 $ linearly independent Lamé functions.

Solutions of equation (*) that are linearly independent with the Lamé functions of the first kind and are obtained by means of the Liouville–Ostrogradski formula are called Lamé functions of the second kind.

For references see Lamé equation.

How to Cite This Entry:
Lamé function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_function&oldid=23368
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article