Ellipsoidal harmonics of the second kind, also known as Lamé functions of the second kind, are variously defined as
![F_m^p(x)=(2m+1)E_m^p(x)int_x^infty(dx)/(sqrt((x^2-b^2)(x^2-c^2))[E_m^p(x)]^2)](/images/equations/EllipsoidalHarmonicoftheSecondKind/NumberedEquation1.svg) |
(Byerly 1959, p. 258) or
![F_m^p(x)=(2m+1)E_m^p(x)int_0^u(du)/([E_m^p(u)]^2)](/images/equations/EllipsoidalHarmonicoftheSecondKind/NumberedEquation2.svg) |
(Heine 1845, p. 194; Whittaker and Watson 1990, p. 562).
Ellipsoidal Harmonic of the Second Kind
See also
Ellipsoidal Harmonic of the First KindExplore with Wolfram|Alpha
References
Byerly, W. E. "Laplace's Equation in Curvilinear Coördinates. Ellipsoidal Harmonics." Ch. 8 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 251-266, 1959.Heine, E. "Beitrag zur Theorie der Anziehung und der Wärme." J. reine angew. Math. 29, 185-208, 1845.Whittaker, E. T. and Watson, G. N. "Lamé Functions of the Second Kind." §23.47 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 562-563, 1990.Referenced on Wolfram|Alpha
Ellipsoidal Harmonic of the Second KindCite this as:
Weisstein, Eric W. "Ellipsoidal Harmonic of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipsoidalHarmonicoftheSecondKind.html