# §29.2 Differential Equations

## §29.2(i) Lamé’s Equation

 29.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu(\nu+1)k^{2}{\operatorname{% sn}}^{2}\left(z,k\right))w=0,$

where $k$ and $\nu$ are real parameters such that $0 and $\nu\geq-\tfrac{1}{2}$. For $\operatorname{sn}\left(z,k\right)$ see §22.2. This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K^{\prime}}$, where $p,q\in\mathbb{Z}$, and $K$, ${K^{\prime}}$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime}(=(1-k^{2})^{1/2})$, 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.

## §29.2(ii) Other Forms

 29.2.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\xi}^{2}}+\frac{1}{2}\left(\frac{1}{\xi}+% \frac{1}{\xi-1}+\frac{1}{\xi-k^{-2}}\right)\frac{\mathrm{d}w}{\mathrm{d}\xi}+% \frac{hk^{-2}-\nu(\nu+1)\xi}{4\xi(\xi-1)(\xi-k^{-2})}w=0,$

where

 29.2.3 $\xi={\operatorname{sn}}^{2}\left(z,k\right).$ ⓘ Defines: $\xi$: variable (locally) Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $z$: complex variable and $k$: real parameter Permalink: http://dlmf.nist.gov/29.2.E3 Encodings: TeX, pMML, png See also: Annotations for §29.2(ii), §29.2 and Ch.29
 29.2.4 $(1-k^{2}{\cos}^{2}\phi)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\phi}^{2}}+k^{2}% \cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(h-\nu(\nu+1)k^{2}{\cos}^{2% }\phi)w=0,$

where

 29.2.5 $\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$ ⓘ Defines: $\phi$: change of variable (locally) Symbols: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter Referenced by: §29.15(i), §29.15(ii), §29.6(i) Permalink: http://dlmf.nist.gov/29.2.E5 Encodings: TeX, pMML, png See also: Annotations for §29.2(ii), §29.2 and Ch.29

For $\operatorname{am}\left(z,k\right)$ see §22.16(i).

Next, let $e_{1},e_{2},e_{3}$ be any real constants that satisfy $e_{1}>e_{2}>e_{3}$ and

 29.2.6 $\displaystyle e_{1}+e_{2}+e_{3}$ $\displaystyle=0,$ $\displaystyle\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}$ $\displaystyle=k^{2}.$ ⓘ Symbols: $k$: real parameter and $e_{j}$: constants Permalink: http://dlmf.nist.gov/29.2.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.2(ii), §29.2 and Ch.29

(These constants are not unique.) Then with

 29.2.7 $\displaystyle g$ $\displaystyle=(e_{1}-e_{3})h+\nu(\nu+1)e_{3},$ ⓘ Defines: $g$ (locally) Symbols: $h$: real parameter, $\nu$: real parameter and $e_{j}$: constants Permalink: http://dlmf.nist.gov/29.2.E7 Encodings: TeX, pMML, png See also: Annotations for §29.2(ii), §29.2 and Ch.29 29.2.8 $\displaystyle\eta$ $\displaystyle=(e_{1}-e_{3})^{-1/2}(z-\mathrm{i}{K^{\prime}}),$ ⓘ Defines: $\eta$ (locally) Symbols: ${K^{\prime}}\left(\NVar{k}\right)$: Legendre’s complementary complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $z$: complex variable, $k$: real parameter and $e_{j}$: constants Permalink: http://dlmf.nist.gov/29.2.E8 Encodings: TeX, pMML, png See also: Annotations for §29.2(ii), §29.2 and Ch.29

we have

 29.2.9 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+(g-\nu(\nu+1)\wp\left(\eta% \right))w=0,$

and

 29.2.10 ${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\frac{1}{2}\left(\frac{1}{% \zeta-e_{1}}+\frac{1}{\zeta-e_{2}}+\frac{1}{\zeta-e_{3}}\right)\frac{\mathrm{d% }w}{\mathrm{d}\zeta}}+\frac{g-\nu(\nu+1)\zeta}{4(\zeta-e_{1})(\zeta-e_{2})(% \zeta-e_{3})}w=0,$

where

 29.2.11 $\zeta=\wp\left(\eta;g_{2},g_{3}\right)=\wp\left(\eta\right),$ ⓘ

with

 29.2.12 $\displaystyle g_{2}$ $\displaystyle=-4(e_{2}e_{3}+e_{3}e_{1}+e_{1}e_{2}),$ $\displaystyle g_{3}$ $\displaystyle=4e_{1}e_{2}e_{3}.$ ⓘ Defines: $g_{2}$ (locally) and $g_{3}$ (locally) Symbols: $e_{j}$: constants Permalink: http://dlmf.nist.gov/29.2.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §29.2(ii), §29.2 and Ch.29

For the Weierstrass function $\wp$ see §23.2(ii).

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