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Lamé's Differential Equation

The ordinary differential equation

(x^2-b^2)(x^2-c^2)(d^2z)/(dx^2)+x(x^2-b^2+x^2-c^2)(dz)/(dx)-[m(m+1)x^2-(b^2+c^2)p]z=0.
(1)

(Byerly 1959, p. 255). The solution is denoted E_m^p(x) and is known as an ellipsoidal harmonic of the first kind, or Lamé function. Whittaker and Watson (1990, pp. 554-555) give the alternative forms

4Delta_lambdad/(dlambda)[Delta_lambda(dLambda)/(dlambda)]=[n(n+1)lambda+C]Lambda
(2)
(d^2Lambda)/(dlambda^2)+[(1/2)/(a^2+lambda)+(1/2)/(b^2+lambda)+(1/2)/(c^2)](dLambda)/(dlambda)
(3)
=([n(n+1)lambda+C]Lambda)/(4Delta_lambda)
(4)
(d^2Lambda)/(du^2)=[n(n+1)P(u)+C-1/3n(n+1)(a^2+b^2+c^2)]Lambda
(5)
(d^2Lambda)/(dz^2)=n(n+1)k^2sn^2(z,k)+ALambda
(6)

(Whittaker and Watson 1990, pp. 554-555; Ward 1987; Zwillinger 1997, p. 124). Here, P is a Weierstrass elliptic function, sn(z,k) is a Jacobi elliptic function, and

Lambda(theta)=product_(q=1)^(m)(theta-theta_q)
(7)
Delta_lambda=sqrt((a^2+lambda)(b^2+lambda)(c^2+lambda))
(8)
A=(C-1/3n(n+1)(a^2+b^2+c^2)+e_3n(n+1))/(e_1-e_3).
(9)

Two other equations named after Lamé are given by

y^('')+1/2[1/(x-a_1)+1/(x-a_2)+1/(x-a_3)]y^'+1/4[(A_0+A_1x)/((x-a_1)(x-a_2)(x-a_3))]y=0
(10)

and

y^('')+1/2[1/x+1/(x-a_2)+1/(x-a_3)]y^'+1/4[((a_2^2+a_3^2)q-p(p+1)x+kappax^2)/(x(x-a_2)(x-a_3))]y=0
(11)

(Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 124).

See also

Ellipsoidal Wave Equation, Lamé's Differential Equation Types, Wangerin Differential Equation

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References

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, 1959.Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961.Ward, R. S. "The Nahm Equations, Finite-Gap Potentials and Lamé Functions." J. Phys. A: Math. Gen. 20, 2679-2683, 1987.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 124, 1997.

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Lamé's Differential Equation

Cite this as:

Weisstein, Eric W. "Lamé's Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LamesDifferentialEquation.html

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