Lamé's Differential Equation

$\begin{displaymath} (x^2-b^2)(x^2-c^2){d^2z\over dx^2}+x(x^2-b^2+x^2-c^2){dz\over dx}-[m(m+1)x^2-(b^2+c^2)p]z=0. \end{displaymath}$

(1)

(Byerly 1959, p. 255). The solution is denoted

and is known as a Lamé Function or an Ellipsoidal Harmonic. Whittaker and Watson (1990, pp. 554-555) give the alternative forms

$\begin{displaymath} 4\Delta_\lambda {d\over d\lambda}\left[{\Delta_\lambda{d\Lambda\over d\lambda}}\right]=[n(n+1)\lambda+C]\Lambda \end{displaymath}$

(2)

$\begin{displaymath} {d^2\Lambda\over d\lambda^2}+\left[{{{\textstyle{1\over 2}}\... ...over d\lambda}={[n(n+1)\lambda+C]\Lambda\over 4\Delta_\lambda} \end{displaymath}$

(3)

$\begin{displaymath} {d^2\Lambda\over du^2}=[n(n+1)\wp(u)+C-{\textstyle{1\over 3}}n(n+1)(a^2+b^2+c^2)]\Lambda \end{displaymath}$

(4)

$\begin{displaymath} {d^2\Lambda\over{dz_1}^2}=[n(n+1)k^2\mathop{\rm sn}\nolimits ^2\alpha+A]\Lambda, \end{displaymath}$

(5)

where $\wp$ is a Weierstraß Elliptic Function and

$\displaystyle \Lambda(\theta)$	$\textstyle \equiv$	$\displaystyle \prod_{q=1}^m(\theta-\theta_q)$	(6)
$\displaystyle \Delta_\lambda$	$\textstyle \equiv$	$\displaystyle \sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)}$	(7)
$\displaystyle A$	$\textstyle \equiv$	$\displaystyle {C-{\textstyle{1\over 3}}n(n+1)(a^2+b^2+c^2)+e_3 n(n+1)\over e_1-e_3}.$
			(8)

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.

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