31 Heun Functions Properties 31.7 Relations to Other Functions 31.9 Orthogonality

§31.8 Solutions via Quadratures

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Keywords:: Hermite–Darboux method, Heun functions, Heun’s equation, Kovacic’s algorithm, Lamé functions, Liouvillean solutions, automorphisms, relations to Heun functions, solutions via quadratures
Permalink:: http://dlmf.nist.gov/31.8
See also:: Annotations for Ch.31

For half-odd-integer values of the exponent parameters:

31.8.1		$\displaystyle\beta-\alpha$	$\displaystyle=m_{0}+\tfrac{1}{2},$
		$\displaystyle\gamma$	$\displaystyle=-m_{1}+\tfrac{1}{2},$
		$\displaystyle\delta$	$\displaystyle=-m_{2}+\tfrac{1}{2},$
		$\displaystyle\epsilon$	$\displaystyle=-m_{3}+\tfrac{1}{2}$ ,
	$m_{0},m_{1},m_{2},m_{3}=0,1,2,\dots$ ,
ⓘ Symbols: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter Referenced by: §31.8 Permalink: http://dlmf.nist.gov/31.8.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §31.8 and Ch.31

the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows.

Denote $\mathbf{m}=(m_{0},m_{1},m_{2},m_{3})$ and $\lambda=-4q$ . Then

31.8.2		$w_{\pm}(\mathbf{m};\lambda;z)=\sqrt{\Psi_{g,N}(\lambda,z)}\*\exp\left(\pm\frac% {i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\,% \mathrm{d}t}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}\right)$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\lambda=-4q$ , $\Psi_{g,N}(\lambda,z)$ : polynomial, $g$ : degree of $\lambda$ and $N$ : degree of $z$ Referenced by: §31.8 Permalink: http://dlmf.nist.gov/31.8.E2 Encodings: TeX, pMML, png See also: Annotations for §31.8 and Ch.31

are two independent solutions of (31.2.1). Here $\Psi_{g,N}(\lambda,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_{0}+m_{1}+m_{2}+m_{3}$ in $z$ , that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. The degree $g$ is given by

31.8.3		$g=\tfrac{1}{2}\max\left(2\max_{0\leq k\leq 3}m_{k},1+N-(1+(-1)^{N})\left(% \tfrac{1}{2}+\min_{0\leq k\leq 3}m_{k}\right)\right).$
ⓘ Symbols: $m$ : nonnegative integer, $g$ : degree of $\lambda$ and $N$ : degree of $z$ Permalink: http://dlmf.nist.gov/31.8.E3 Encodings: TeX, pMML, png See also: Annotations for §31.8 and Ch.31

The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\Gamma:\nu^{2}=\prod_{j=1}^{2g+1}(\lambda-\lambda_{j})$ . (This $\nu$ is unrelated to the $\nu$ in §31.6.) Lastly, $\lambda_{j}$ , $j=1,2,\ldots,2g+1$ , are the zeros of the Wronskian of $w_{+}(\mathbf{m};\lambda;z)$ and $w_{-}(\mathbf{m};\lambda;z)$ .

By automorphisms from §31.2(v), similar solutions also exist for $m_{0},m_{1},m_{2},m_{3}\in\mathbb{Z}$ , and $\Psi_{g,N}(\lambda,z)$ may become a rational function in $z$ . For instance,

31.8.4		$\displaystyle\Psi_{1,2}$	$\displaystyle=z^{2}+\lambda z+a,$
		$\displaystyle\nu^{2}$	$\displaystyle=(\lambda+a+1)(\lambda^{2}-4a)$ ,
	$\mathbf{m}=(1,1,0,0)$ ,
ⓘ Symbols: $z$ : complex variable, $\nu$ : real or complex parameter, $a$ : complex parameter, $\lambda=-4q$ and $\Psi_{g,N}(\lambda,z)$ : polynomial Permalink: http://dlmf.nist.gov/31.8.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.8 and Ch.31

and

31.8.5		$\displaystyle\Psi_{1,-1}$	$\displaystyle=\left(z^{2}+(\lambda+3a+3)z+a\right)/z^{3},$
		$\displaystyle\nu^{2}$	$\displaystyle=(\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a\right)$ ,
	$\mathbf{m}=(1,-2,0,0)$ .
ⓘ Symbols: $z$ : complex variable, $\nu$ : real or complex parameter, $a$ : complex parameter, $\lambda=-4q$ and $\Psi_{g,N}(\lambda,z)$ : polynomial Referenced by: Erratum (V1.0.7) for Equation (31.8.5) Permalink: http://dlmf.nist.gov/31.8.E5 Encodings: TeX, TeX, pMML, pMML, png, png Errata (effective with 1.0.7): Originally the first term on the right side of the equation for $\Psi_{1,-1}$ was $z^{3}$ . The correct term is $z^{2}$ . Reported 2013-07-25 by Christopher Künstler See also: Annotations for §31.8 and Ch.31

For $\mathbf{m}=(m_{0},0,0,0)$ , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. The curve $\Gamma$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_{j}\in\mathbb{Z}$ . When $\lambda=-4q$ approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. For more details see Smirnov (2002).

The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).

31.7 Relations to Other Functions 31.9 Orthogonality

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