What's New
About the Project
NIST
29 Lamé FunctionsLamé Functions

§29.2 Differential Equations

Contents

§29.2(i) Lamé’s Equation

29.2.1 d2wdz2+(h-ν(ν+1)k2sn2(z,k))w=0,

where k and ν are real parameters such that 0<k<1 and ν-12. For sn(z,k) see §22.2. This equation has regular singularities at the points 2pK+(2q+1)iK, where p,q, and K, K are the complete elliptic integrals of the first kind with moduli k, k(=(1-k2)1/2), respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). See Figure 29.2.1.

\begin{picture}(10.0,8.0)(-5.0,-4.0)\put(0.0,-4.0){\line(0,1){8.0}}
\put(-5.0,0.0){\line(1,0){10.0}}
\put(-4.17,-3.15){$\times$}\put(-2.17,-3.15){$\times$}\put(-0.17,-3.15){$%
\times$}\put(1.83,-3.15){$\times$}\put(3.83,-3.15){$\times$}
\put(-4.17,-1.15){$\times$}\put(-2.17,-1.15){$\times$}\put(-0.17,-1.15){$%
\times$}\put(1.83,-1.15){$\times$}\put(3.83,-1.15){$\times$}
\put(-4.17,0.85){$\times$}\put(-2.17,0.85){$\times$}\put(-0.17,0.85){$\times$}%
\put(1.83,0.85){$\times$}\put(3.83,0.85){$\times$}
\put(-4.17,2.85){$\times$}\put(-2.17,2.85){$\times$}\put(-0.17,2.85){$\times$}%
\put(1.83,2.85){$\times$}\put(3.83,2.85){$\times$}
{\put(-4.0,0.0){\line(0,1){0.15}}\put(-2.0,0.0){\line(0,1){0.15}}\put(0.0,0.0)%
{\line(0,1){0.15}}\put(2.0,0.0){\line(0,1){0.15}}\put(4.0,0.0){\line(0,1){0.15%
}}}
{\put(0.0,-2.0){\line(1,0){0.15}}\put(0.0,2.0){\line(1,0){0.15}}}
\put(0.1,0.1){0}
\put(-1.5,-3.15){$-3\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}%
}\/}\nolimits}\!$}
\put(-1.5,-2.15){$-2\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}%
}\/}\nolimits}\!$}
\put(-1.3,-1.15){$-\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}}%
\/}\nolimits}\!$}
\put(-0.9,0.85){$\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}}\/%
}\nolimits}\!$}
\put(-1.1,1.85){$2\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}}%
\/}\nolimits}\!$}
\put(-1.1,2.85){$3\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{\prime}}%
\/}\nolimits}\!$}
\put(-4.6,-0.5){$-4\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(-2.6,-0.5){$-2\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(1.8,-0.5){$2\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(3.8,-0.5){$4\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\end{picture}
Figure 29.2.1: z-plane: singularities ××× of Lamé’s equation. Magnify

§29.2(ii) Other Forms

29.2.2 d2wdξ2+12(1ξ+1ξ-1+1ξ-k-2)dwdξ+hk-2-ν(ν+1)ξ4ξ(ξ-1)(ξ-k-2)w=0,

where

29.2.3 ξ=sn2(z,k).
29.2.4 (1-k2cos2ϕ)d2wdϕ2+k2cosϕsinϕdwdϕ+(h-ν(ν+1)k2cos2ϕ)w=0,

where

29.2.5 ϕ=12π-am(z,k).

For am(z,k) see §22.16(i).

Next, let e1,e2,e3 be any real constants that satisfy e1>e2>e3 and

29.2.6 e1+e2+e3 =0,
(e2-e3)/(e1-e3) =k2.

(These constants are not unique.) Then with

29.2.7 g =(e1-e3)h+ν(ν+1)e3,
29.2.8 η =(e1-e3)-1/2(z-iK),

we have

29.2.9 d2wdη2+(g-ν(ν+1)(η))w=0,

and

29.2.10 d2wdζ2+12(1ζ-e1+1ζ-e2+1ζ-e3)dwdζ+g-ν(ν+1)ζ4(ζ-e1)(ζ-e2)(ζ-e3)w=0,

where

29.2.11 ζ=(η;g2,g3)=(η),

with

29.2.12 g2 =-4(e2e3+e3e1+e1e2),
g3 =4e1e2e3.

For the Weierstrass function see §23.2(ii).

Equation (29.2.10) is a special case of Heun’s equation (31.2.1).