Ellipsoidal harmonic

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.

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