TOPICS
Search

Lamé's Differential Equation

Lamé's differential equation is a second-order linear ordinary differential equation. In one form it is

(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+lambda)](dLambda)/(dlambda)
(3)
=([n(n+1)lambda+C]Lambda)/(4Delta_lambda^2)
(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)+A]Lambda
(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)

In the Jacobi form above, u=sn^2(z,k) transforms the equation to Heun's differential equation. For k^2!=0,1, the resulting parameters in Heun's notation are a=1/k^2, q=A/(4k^2), alpha=-n/2, beta=(n+1)/2, and gamma=delta=epsilon=1/2 (cf. NIST DLMF, §29.2(ii)). The Jacobi form is also used to define the periodic Lamé functions.

Two other equations named after Lamé, with primes denoting derivatives of y(x) with respect to x, are

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, Heun's Differential Equation, Lamé Functions, Lamé's Differential Equation Types, Wangerin Differential Equation

Explore with Wolfram|Alpha

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.National Institute of Standards and Technology. "Lamé's Equation." §29.2 in Digital Library of Mathematical Functions. https://dlmf.nist.gov/29.2.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.

Referenced on Wolfram|Alpha

Lamé's Differential Equation

Cite this as:

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

Subject classifications