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§29.2(ii) Other Forms

ⓘ

Keywords:: Heun’s equation, Jacobian elliptic-function form, Lamé’s equation, Weierstrass elliptic-function form, other forms, relation to Heun’s equation, relation to Lamé’s equation, trigonometric form
Notes:: See Erdélyi et al. (1955, §15.2).
Permalink:: http://dlmf.nist.gov/29.2.ii
See also:: Annotations for §29.2 and Ch.29

29.2.2		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\xi}^{2}}+\frac{1}{2}\left(\frac{1}{\xi}+% \frac{1}{\xi-1}+\frac{1}{\xi-k^{-2}}\right)\frac{\mathrm{d}w}{\mathrm{d}\xi}+% \frac{hk^{-2}-\nu(\nu+1)\xi}{4\xi(\xi-1)(\xi-k^{-2})}w=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §29.2(ii), §29.2 and Ch.29

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		$(1-k^{2}{\cos}^{2}\phi)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\phi}^{2}}+k^{2}% \cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(h-\nu(\nu+1)k^{2}{\cos}^{2% }\phi)w=0,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : 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 See also: Annotations for §29.2(ii), §29.2 and Ch.29

where

29.2.5		$\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$
ⓘ 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	$\displaystyle e_{1}+e_{2}+e_{3}$	$\displaystyle=0,$
29.2.6	$\displaystyle\ifrac{(e_{2}-e_{3})}{(e_{1}-e_{3})}$	$\displaystyle=k^{2}.$
ⓘ 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		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\eta}^{2}}+(g-\nu(\nu+1)\wp\left(\eta% \right))w=0,$
ⓘ Symbols: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z\|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §29.2(ii), §29.2 and Ch.29

and

29.2.10		${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\frac{1}{2}\left(\frac{1}{% \zeta-e_{1}}+\frac{1}{\zeta-e_{2}}+\frac{1}{\zeta-e_{3}}\right)\frac{\mathrm{d% }w}{\mathrm{d}\zeta}}+\frac{g-\nu(\nu+1)\zeta}{4(\zeta-e_{1})(\zeta-e_{2})(% \zeta-e_{3})}w=0,$
ⓘ 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		$\zeta=\wp\left(\eta;g_{2},g_{3}\right)=\wp\left(\eta\right),$
ⓘ 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	$\displaystyle g_{2}$	$\displaystyle=-4(e_{2}e_{3}+e_{3}e_{1}+e_{1}e_{2}),$
29.2.12	$\displaystyle g_{3}$	$\displaystyle=4e_{1}e_{2}e_{3}.$
ⓘ 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).

29.2.1		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(h-\nu(\nu+1)k^{2}{\operatorname{% sn}}^{2}\left(z,k\right))w=0,$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §29.2(i), §29.2 and Ch.29