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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 four types as

$\displaystyle 4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right]$ $\textstyle =$ $\displaystyle [2m(2m+1)\theta+C]\Lambda(\theta)$  
      (1)
$\displaystyle 4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right]$ $\textstyle =$ $\displaystyle [(2m+1)(2m+2)\theta+C]\Lambda(\theta)$  
      (2)
$\displaystyle 4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right]$ $\textstyle =$ $\displaystyle [(2m+2)(2m+3)\theta+C]\Lambda(\theta)$  
      (3)
$\displaystyle 4\Delta(\theta) {d\over d\theta}\left[{F(\theta){d\Lambda(\theta)\over d\theta}}\right]$ $\textstyle =$ $\displaystyle [(2m+3)(2m+4)\theta+C]\Lambda(\theta),$  
      (4)

where
$\displaystyle \Delta(\theta)$ $\textstyle \equiv$ $\displaystyle \sqrt{(a^2+\theta)(b^2+\theta)(c^2+\theta)}$ (5)
$\displaystyle \Lambda(\theta)$ $\textstyle \equiv$ $\displaystyle \prod_{q=1}^m(\theta-\theta_q).$ (6)


References

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




© 1996-9 Eric W. Weisstein
1999-05-26