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Lamé's Differential Equation Types

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as

4delta(theta)d/(dtheta)[f(theta)(dlambda(theta))/(dtheta)]=[2m(2m+1)theta+c]lambda(theta)
(1)
4delta(theta)d/(dtheta)[f(theta)(dlambda(theta))/(dtheta)]=[(2m+1)(2m+2)theta+c]lambda(theta)
(2)
4delta(theta)d/(dtheta)[f(theta)(dlambda(theta))/(dtheta)]=[(2m+2)(2m+3)theta+c]lambda(theta)
(3)
4delta(theta)d/(dtheta)[f(theta)(dlambda(theta))/(dtheta)]=[(2m+3)(2m+4)theta+c]lambda(theta),
(4)

where

delta(theta)=sqrt((a^2+theta)(b^2+theta)(c^2+theta))
(5)
lambda(theta)=product_(q=1)^(m)(theta-theta_q).
(6)

See also

Lamé's Differential Equation

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References

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Lamé's Differential Equation Types

Cite this as:

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

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