Let be Cartesian coordinates in the Euclidean space , related to the ellipsoidal coordinates by three formulas of the same form
where , and . Putting , one obtains coordinate surfaces in the form of ellipsoids. A harmonic function that is a solution of the Laplace equation can be written as a linear combination of expressions of the form
where the factors , , are solutions of the Lamé equation. Expressions of the form (*) for and their linear combinations are called ellipsoidal harmonics or, better, surface ellipsoidal harmonics, in contrast to combinations of expressions (*) depending on all three variables , which are sometimes called spatial ellipsoidal harmonics.
|||A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)|
|||P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1–2 , McGraw-Hill (1953)|
Ellipsoidal harmonic. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Ellipsoidal_harmonic&oldid=11263