Ellipsoidal harmonics of the second kind, also known as Lamé functions of the second kind, are variously defined as
(Byerly 1959, p. 258). In the elliptic parametrization 
,
the corresponding expression is
where 
is the Weierstrass
elliptic function (Heine 1845, p. 194; Whittaker and Watson 1990, p. 562).
See also
Ellipsoidal Harmonic of the First Kind,
Weierstrass Elliptic
Function
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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 Kind
Cite this as:
Weisstein, Eric W. "Ellipsoidal Harmonic of the Second Kind." From MathWorld--A Wolfram Resource.
https://mathworld.wolfram.com/EllipsoidalHarmonicoftheSecondKind.html
Subject classifications
![F_m^p(x)=(2m+1)E_m^p(x)int_x^infty(dt)/(sqrt((t^2-b^2)(t^2-c^2))[E_m^p(t)]^2)](/images/equations/EllipsoidalHarmonicoftheSecondKind/NumberedEquation1.svg)
![F_m^p(xi)=(2m+1)E_m^p(xi)int_0^u(dv)/([E_m^p(P(v))]^2),](/images/equations/EllipsoidalHarmonicoftheSecondKind/NumberedEquation2.svg)
