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29 Lamé FunctionsLamé Functions

§29.2 Differential Equations

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Permalink:
http://dlmf.nist.gov/29.2
See also:
Annotations for Ch.29
Contents
  1. §29.2(i) Lamé’s Equation
  2. §29.2(ii) Other Forms

§29.2(i) Lamé’s Equation

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Keywords:
Lamé’s equation, algebraic form, singularities
Notes:
See Erdélyi et al. (1955, §15.2).
Permalink:
http://dlmf.nist.gov/29.2.i
See also:
Annotations for §29.2 and Ch.29

where k and \nu are real parameters such that 0<k<1 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).
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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

where

29.2.5 \phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).
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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
e_{1}+e_{2}+e_{3}=0,
\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}=k^{2}.
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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 g=(e_{1}-e_{3})h+\nu(\nu+1)e_{3},
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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

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,
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Symbols:
\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}: derivative of f with respect to x, \nu: real parameter, e_{j}: constants, g and \zeta
Referenced by:
§29.2(ii)
Permalink:
http://dlmf.nist.gov/29.2.E10
Encodings:
TeX, pMML, png
See also:
Annotations for §29.2(ii), §29.2 and Ch.29

where

29.2.11 \zeta=\wp\left(\eta;g_{2},g_{3}\right)=\wp\left(\eta\right),
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Defines:
\zeta (locally)
Symbols:
\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right): Weierstrass \wp-function, \wp\left(\NVar{z}\right) (= \wp\left(z|\mathbb{L}\right) = \wp\left(z;g_{2},g_{3}\right)): Weierstrass \wp-function, \eta, g_{2} and g_{3}
Permalink:
http://dlmf.nist.gov/29.2.E11
Encodings:
TeX, pMML, png
See also:
Annotations for §29.2(ii), §29.2 and Ch.29

with

29.2.12
g_{2}=-4(e_{2}e_{3}+e_{3}e_{1}+e_{1}e_{2}),
g_{3}=4e_{1}e_{2}e_{3}.
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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).

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