31 Heun Functions Properties 31.13 Asymptotic Approximations 31.15 Stieltjes Polynomials

§31.14 General Fuchsian Equation

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Permalink:: http://dlmf.nist.gov/31.14
See also:: Annotations for Ch.31

§31.14(i) Definitions

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Keywords:: Fuchsian equation, classification of parameters, definitions
Notes:: See Ince (1926, Chapter XV).
Permalink:: http://dlmf.nist.gov/31.14.i
See also:: Annotations for §31.14 and Ch.31

The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_{j}$ , $j=1,2,\dots,N$ , and at $\infty$ , is given by

31.14.1		${\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},$
$\sum_{j=1}^{N}q_{j}=0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities Referenced by: §31.15(i) Permalink: http://dlmf.nist.gov/31.14.E1 Encodings: TeX, pMML, png See also: Annotations for §31.14(i), §31.14 and Ch.31

The exponents at the finite singularities $a_{j}$ are $\{0,{1-\gamma_{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$ , where

31.14.2	$\displaystyle\alpha+\beta+1$	$\displaystyle=\sum_{j=1}^{N}\gamma_{j},$
31.14.2	$\displaystyle\alpha\beta$	$\displaystyle=\sum_{j=1}^{N}a_{j}q_{j}.$
ⓘ Symbols: $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $N+1$ : number of singularities Permalink: http://dlmf.nist.gov/31.14.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.14(i), §31.14 and Ch.31

The three sets of parameters comprise the singularity parameters $a_{j}$ , the exponent parameters $\alpha,\beta,\gamma_{j}$ , and the $N-2$ free accessory parameters $q_{j}$ . With $a_{1}=0$ and $a_{2}=1$ the total number of free parameters is $3N-3$ . Heun’s equation (31.2.1) corresponds to $N=3$ .

Normal Form

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Keywords:: Fuchsian equation, Heun’s equation, normal form, relation to Fuchsian equation, relation to Heun’s equation
See also:: Annotations for §31.14(i), §31.14 and Ch.31

31.14.3		$w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
ⓘ Symbols: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution Permalink: http://dlmf.nist.gov/31.14.E3 Encodings: TeX, pMML, png See also: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

31.14.4		$\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,$
$\sum_{j=1}^{N}\tilde{q}_{j}=0$ ,
ⓘ Defines: $W(z)$ : solution (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities Permalink: http://dlmf.nist.gov/31.14.E4 Encodings: TeX, pMML, png See also: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

31.14.5	$\displaystyle\tilde{q}_{j}$	$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{N}\frac{\gamma_{j}\gamma_{k}}{a_{j}-a_{k}}-q_{j},$
31.14.5	$\displaystyle\tilde{\gamma}_{j}$	$\displaystyle=\frac{\gamma_{j}}{2}\left(\frac{\gamma_{j}}{2}-1\right).$
ⓘ Symbols: $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities Permalink: http://dlmf.nist.gov/31.14.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

§31.14(ii) Kovacic’s Algorithm

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Keywords:: Heun functions, Kovacic’s algorithm, spatio-temporal dynamics
Referenced by:: §31.8
Permalink:: http://dlmf.nist.gov/31.14.ii
See also:: Annotations for §31.14 and Ch.31

An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). The algorithm returns a list of solutions if they exist.

For applications of Kovacic’s algorithm in spatio-temporal dynamics see Rod and Sleeman (1995).

31.13 Asymptotic Approximations 31.15 Stieltjes Polynomials

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§31.14 General Fuchsian Equation

Contents

§31.14(i) Definitions

Normal Form

§31.14(ii) Kovacic’s Algorithm