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A function of a point on an ellipsoid that appears in the solution of the [[Laplace equation|Laplace equation]] by the method of separation of variables in [[Ellipsoidal coordinates|ellipsoidal coordinates]].
 
A function of a point on an ellipsoid that appears in the solution of the [[Laplace equation|Laplace equation]] by the method of separation of variables in [[Ellipsoidal coordinates|ellipsoidal coordinates]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354301.png" /> be Cartesian coordinates in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354302.png" />, related to the ellipsoidal coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354303.png" /> by three formulas of the same form
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Let $  ( x , y , z ) $
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be Cartesian coordinates in the Euclidean space $  \mathbf R  ^ {3} $,  
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related to the ellipsoidal coordinates $  ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $
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by three formulas of the same form
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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/e/e035/e035430/e0354304.png" /></td> </tr></table>
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\frac{x  ^ {2}}{ {\xi _ 1}  ^ {2} - a  ^ {2} }
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+
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\frac{y  ^ {2}}{ {\xi _ 2}  ^ {2} - b  ^ {2} }
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+
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\frac{z  ^ {2}}{ {\xi _ 3}  ^ {2} - c  ^ {2} }
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  = 1 ,\ \
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> > > 0 ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354307.png" />. Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354308.png" />, one obtains coordinate surfaces in the form of ellipsoids. A harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e0354309.png" /> that is a solution of the Laplace equation can be written as a linear combination of expressions of the form
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where $  a < \xi _ {1} < + \infty $,  
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$  b < \xi _ {2} < a $
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and $  c < \xi _ {3} < b $.  
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Putting $  \xi _ {1} = \xi _ {1}  ^ {0} $,  
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one obtains coordinate surfaces in the form of ellipsoids. A harmonic function $  h = h ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $
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that is a solution of the Laplace equation can be written as a linear combination of expressions of the 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/e/e035/e035430/e03543010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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E _ {1} ( \xi _ {1} ) E _ {2} ( \xi _ {2} ) E _ {3} ( \xi _ {3} ) ,
 +
$$
  
where the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e03543011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e03543012.png" />, are solutions of the [[Lamé equation|Lamé equation]]. Expressions of the form (*) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e03543013.png" /> and their linear combinations are called ellipsoidal harmonics or, better, surface ellipsoidal harmonics, in contrast to combinations of expressions (*) depending on all three variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035430/e03543014.png" />, which are sometimes called spatial ellipsoidal harmonics.
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where the factors $  E _ {j} ( \xi _ {j} ) $,
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$  j = 1 , 2 , 3 $,  
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are solutions of the [[Lamé equation|Lamé equation]]. Expressions of the form (*) for $  \xi _ {1} = \xi _ {1}  ^ {0} $
 +
and their linear combinations are called ellipsoidal harmonics or, better, surface ellipsoidal harmonics, in contrast to combinations of expressions (*) depending on all three variables $  ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $,  
 +
which are sometimes called spatial ellipsoidal harmonics.
  
 
====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>

Latest revision as of 19:37, 5 June 2020


A function of a point on an ellipsoid that appears in the solution of the Laplace equation by the method of separation of variables in ellipsoidal coordinates.

Let $ ( x , y , z ) $ be Cartesian coordinates in the Euclidean space $ \mathbf R ^ {3} $, related to the ellipsoidal coordinates $ ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $ by three formulas of the same form

$$ \frac{x ^ {2}}{ {\xi _ 1} ^ {2} - a ^ {2} } + \frac{y ^ {2}}{ {\xi _ 2} ^ {2} - b ^ {2} } + \frac{z ^ {2}}{ {\xi _ 3} ^ {2} - c ^ {2} } = 1 ,\ \ a > b > c > 0 , $$

where $ a < \xi _ {1} < + \infty $, $ b < \xi _ {2} < a $ and $ c < \xi _ {3} < b $. Putting $ \xi _ {1} = \xi _ {1} ^ {0} $, one obtains coordinate surfaces in the form of ellipsoids. A harmonic function $ h = h ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $ that is a solution of the Laplace equation can be written as a linear combination of expressions of the form

$$ \tag{* } E _ {1} ( \xi _ {1} ) E _ {2} ( \xi _ {2} ) E _ {3} ( \xi _ {3} ) , $$

where the factors $ E _ {j} ( \xi _ {j} ) $, $ j = 1 , 2 , 3 $, are solutions of the Lamé equation. Expressions of the form (*) for $ \xi _ {1} = \xi _ {1} ^ {0} $ and their linear combinations are called ellipsoidal harmonics or, better, surface ellipsoidal harmonics, in contrast to combinations of expressions (*) depending on all three variables $ ( \xi _ {1} , \xi _ {2} , \xi _ {3} ) $, which are sometimes called spatial ellipsoidal harmonics.

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)
How to Cite This Entry:
Ellipsoidal harmonic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ellipsoidal_harmonic&oldid=46807
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article