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Lamé's Differential Equation

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

{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)

{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)

{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)
$\displaystyle A$ $\textstyle \equiv$ $\displaystyle {C-{\textstyle{1\over 3}}n(n+1)(a^2+b^2+c^2)+e_3 n(n+1)\over e_1-e_3}.$  


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.

© 1996-9 Eric W. Weisstein