§29.2 Differential Equations

Permalink:: http://dlmf.nist.gov/29.2

§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms

§29.2(i) Lamé’s Equation

Notes:: See Erdélyi et al. (1955, §15.2).
Keywords:: Lamé’s equation, singularities
Permalink:: http://dlmf.nist.gov/29.2.i

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

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution
Referenced by:: §29.10, §29.11, §29.12(i), §29.17(i), §29.17(ii), §29.17(iii), §29.18(i), §29.2(i), §29.20(i), §29.3(i), §29.7(ii), §29.8, §29.9
Permalink:: http://dlmf.nist.gov/29.2.E1
Encodings:: TeX, pMML, png

where $k$ and $\nu$ are real parameters such that $0<k<1$ and $\nu\geq-\tfrac{1}{2}$ . For $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ see §22.2. This equation has regular singularities at the points $2p\!\mathop{K\/}\nolimits\!+(2q+1)i\!\mathop{{K^{{\prime}}}\/}\nolimits\!$ , where $p,q\in\Integer$ , and $\!\mathop{K\/}\nolimits\!$ , $\!\mathop{{K^{{\prime}}}\/}\nolimits\!$ 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.

$\begin{picture}(10.0,8.0)(-5.0,-4.0)\put(0.0,-4.0){\line(0,1){8.0}}\put(-5.0,0.0){\line(1,0){10.0}}\put(-4.17,-3.15){$\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 $\times$ $\times$ $\times$ of Lamé’s equation.

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex variable and $k$ : real parameter
Referenced by:: §29.2(i), §31.7(ii)
Permalink:: http://dlmf.nist.gov/29.2.F1
Encodings:: TeX, png

§29.2(ii) Other Forms

Notes:: See Erdélyi et al. (1955, §15.2).
Keywords:: Heun’s equation, Lamé’s equation
Permalink:: http://dlmf.nist.gov/29.2.ii

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $\xi$ : variable
Referenced by:: §29.12(ii)
Permalink:: http://dlmf.nist.gov/29.2.E2
Encodings:: TeX, pMML, png

where

29.2.3 $\xi={\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right).$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $z$ : complex variable, $k$ : real parameter and $\xi$ : variable
Permalink:: http://dlmf.nist.gov/29.2.E3
Encodings:: TeX, pMML, png

29.2.4 $(1-k^{2}{\mathop{\cos\/}\nolimits^{{2}}}\phi)\frac{{d}^{2}w}{{d\phi}^{2}}+k^{2}\mathop{\cos\/}\nolimits\phi\mathop{\sin\/}\nolimits\phi\frac{dw}{d\phi}+(h-\nu(\nu+1)k^{2}{\mathop{\cos\/}\nolimits^{{2}}}\phi)w=0,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\sin\/}\nolimits z$ : sine function, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $\phi$ : change of variable
Permalink:: http://dlmf.nist.gov/29.2.E4
Encodings:: TeX, pMML, png

where

29.2.5 $\phi=\tfrac{1}{2}\pi-\mathop{\mathrm{am}\/}\nolimits\left(z,k\right).$

Symbols:: $\mathop{\mathrm{am}\/}\nolimits\left(x,k\right)$ : Jacobi’s amplitude function, $z$ : complex variable, $k$ : real parameter and $\phi$ : change of variable
Referenced by:: ¶ ‣ §29.15(i), §29.15(ii), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.2.E5
Encodings:: TeX, pMML, png

For $\mathop{\mathrm{am}\/}\nolimits\left(z,k\right)$ 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

$e_{1}+e_{2}+e_{3}=0,$

$\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}=k^{2}.$

Symbols:: $k$ : real parameter and $e_{j}$ : constants
Permalink:: http://dlmf.nist.gov/29.2.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

(These constants are not unique.) Then with

29.2.7 $g=(e_{{1}}-e_{{3}})h+\nu(\nu+1)e_{{3}},$

Symbols:: $h$ : real parameter, $\nu$ : real parameter, $e_{j}$ : constants and $g$
Permalink:: http://dlmf.nist.gov/29.2.E7
Encodings:: TeX, pMML, png

29.2.8 $\eta=(e_{{1}}-e_{{3}})^{{-1/2}}(z-i\!\mathop{{K^{{\prime}}}\/}\nolimits\!),$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $z$ : complex variable, $k$ : real parameter, $e_{j}$ : constants and $\eta$
Permalink:: http://dlmf.nist.gov/29.2.E8
Encodings:: TeX, pMML, png

we have

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

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\nu$ : real parameter, $g$ and $\eta$
Permalink:: http://dlmf.nist.gov/29.2.E9
Encodings:: TeX, pMML, png

and

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