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Ellipsoidal Harmonic of the First Kind

The first solution to Lamé's differential equation, denoted E_n^m(x) for m=1, ..., 2n+1. They are also called Lamé functions of the first kind. The product of two ellipsoidal harmonics of the first kind is a spherical harmonic. Whittaker and Watson (1990, pp. 536-537) write

Theta_p=(x^2)/(a^2+theta_p)+(y^2)/(b^2+theta_p)+(z^2)/(c^2+theta_p)-1
(1)
Pi(Theta)=Theta_1Theta_2...Theta_m,
(2)

and give various types of ellipsoidal harmonics and their highest degree terms as

1. Pi(Theta):2m

2. xPi(Theta),yPi(Theta),zPi(Theta):2m+1

3. yzPi(Theta),zxPi(Theta),xyPi(Theta):2m+2

4. xyzPi(Theta):2m+3.

A Lamé function of degree n may be expressed as

(theta+a^2)^(kappa_1)(theta+b^2)^(kappa_2)(theta+c^2)^(kappa_3)product_(p=1)^m(theta-theta_p),
(3)

where kappa_i=0 or 1/2, theta_i are real and unequal to each other and to -a^2, -b^2, and -c^2, and

1/2n=m+kappa_1+kappa_2+kappa_3.
(4)

Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by K(x), L(x), M(x), and N(x),

K_0(x)=1
(5)
L_0(x)=0
(6)
M_0(x)=0
(7)
N_0(x)=0
(8)
K_1(x)=x
(9)
L_1(x)=sqrt(x^2-b^2)
(10)
M_1(x)=sqrt(x^2-c^2)
(11)
N_1(x)=0
(12)
K_2^(p_1)(x)=x^2-1/3[b^2+c^2-sqrt((b^2+c^2)^2-3b^2c^2)]
(13)
K_2^(p_2)(x)=x^2-1/3[b^2+c^2+sqrt((b^2+c^2)^2-3b^2c^2)]
(14)
L_2(x)=xsqrt(x^2-b^2)
(15)
M_2(x)=xsqrt(x^2-c^2)
(16)
N_2(x)=sqrt((x^2-b^2)(x^2-c^2))
(17)
K_3^(p_1)(x)=x^3-1/5x[2(b^2+c^2)-sqrt(4(b^2+c^2)^2-15b^2c^2)]
(18)
K_3^(p_2)(x)=x^3-1/5x[2(b^2+c^2)+sqrt(4(b^2+c^2)^2-15b^2c^2)]
(19)
L_3^(q_1)(x)=sqrt(x^2-b^2)[x^2-1/5(b^2+2c^2-sqrt((b^2+2c^2)^2-5b^2c^2))]
(20)
L_3^(q_2)(x)=sqrt(x^2-b^2)[x^2-1/5(b^2+2c^2+sqrt((b^2+2c^2)^2-5b^2c^2))]
(21)
M_3^(q_1)(x)=sqrt(x^2-c^2)[x^2-1/5(2b^2+c^2-sqrt((2b^2+c^2)^2-5b^2c^2))]
(22)
M_3^(q_2)(x)=sqrt(x^2-c^2)[x^2-1/5(2b^2+c^2+sqrt((2b^2+c^2)^2-5b^2c^2))]
(23)
M_3^(q_3)(x)=xsqrt((x^2-b^2)(x^2-c^2)).
(24)

See also

Ellipsoidal Harmonic of the Second Kind, Stieltjes' Theorem

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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.Humbert, P. Fonctions de Lamé et Fonctions de Mathieu. Paris: Gauthier-Villars, 1926.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Ellipsoidal Harmonic of the First Kind

Cite this as:

Weisstein, Eric W. "Ellipsoidal Harmonic of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipsoidalHarmonicoftheFirstKind.html

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