§31.8 Solutions via Quadratures

Keywords:: Hermite–Darboux method, Heun functions, Heun’s equation, Kovacic’s algorithm, Lamé functions
Permalink:: http://dlmf.nist.gov/31.8

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

31.8.1

$\beta-\alpha=m_{0}+\tfrac{1}{2},$

$\gamma=-m_{1}+\tfrac{1}{2},$

$\delta=-m_{2}+\tfrac{1}{2},$

$\epsilon=-m_{3}+\tfrac{1}{2}$ , $m_{0},m_{1},m_{2},m_{3}=0,1,2,\dots$ ,

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
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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 $w_{{\pm}}(\mathbf{m};\lambda;z)=\sqrt{\Psi _{{g,N}}(\lambda,z)}\*\mathop{\exp\/}\nolimits\!\left(\pm\frac{i\nu(\lambda)}{2}\int _{{z_{0}}}^{z}\frac{t^{{m_{1}}}(t-1)^{{m_{2}}}(t-a)^{{m_{3}}}dt}{\Psi _{{g,N}}(\lambda,t)\sqrt{t(t-1)(t-a)}}\right)$

Symbols:: $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : 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

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:: $(a,b)$ : open interval, $m$ : nonnegative integer, $g$ : degree of $\lambda$ and $N$ : degree of $z$
Permalink:: http://dlmf.nist.gov/31.8.E3
Encodings:: TeX, pMML, png

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 $m_{0},m_{1},m_{2},m_{3}\in\Integer$ , and $\Psi _{{g,N}}(\lambda,z)$ may become a rational function in $z$ . For instance,

31.8.4

$\Psi _{{1,2}}=z^{2}+\lambda z+a,$

$\nu^{2}=(\lambda+a+1)(\lambda^{2}-4a)$ , $\mathbf{m}=(1,1,0,0)$ ,

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
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and

31.8.5

$\Psi _{{1,-1}}=\left(z^{3}+(\lambda+3a+3)z+a\right)/z^{3},$

$\nu^{2}=(\lambda+4a+4)\left((\lambda+3a+3)^{2}-4a\right)$ , $\mathbf{m}=(1,-2,0,0)$ .

Symbols:: $z$ : complex variable, $\nu$ : real or complex parameter, $a$ : complex parameter, $\lambda=-4q$ and $\Psi _{{g,N}}(\lambda,z)$ : polynomial
Permalink:: http://dlmf.nist.gov/31.8.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

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 $m_{j}\in\Integer$ . 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).