§31.14 General Fuchsian Equation

Permalink:: http://dlmf.nist.gov/31.14

§31.14(i) Definitions
§31.14(ii) Kovacic’s Algorithm

§31.14(i) Definitions

Notes:: See Ince (1926, Chapter XV).
Keywords:: Fuchsian equation
Permalink:: http://dlmf.nist.gov/31.14.i

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 ${\frac{{d}^{2}w}{{dz}^{2}}+\left(\sum _{{j=1}}^{N}\frac{\gamma _{j}}{z-a_{j}}\right)\frac{dw}{dz}+\left(\sum _{{j=1}}^{N}\frac{q_{j}}{z-a_{j}}\right)w=0},$ $\sum _{{j=1}}^{N}q_{j}=0$ .

Symbols:: $\frac{df}{dx}$ : 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
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The exponents at the finite singularities $a_{j}$ are $\{ 0,{1-\gamma _{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$ , where

31.14.2

$\alpha+\beta+1=\sum _{{j=1}}^{N}\gamma _{j},$

$\alpha\beta=\sum _{{j=1}}^{N}a_{j}q_{j}.$

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
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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

Keywords:: Fuchsian equation, Heun’s equation

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
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31.14.4 $\frac{{d}^{2}W}{{dz}^{2}}=\sum _{{j=1}}^{N}\left(\frac{\tilde{\gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,$ $\sum _{{j=1}}^{N}\tilde{q}_{j}=0$ ,

Symbols:: $\frac{df}{dx}$ : 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, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E4
Encodings:: TeX, pMML, png

31.14.5

$\tilde{q}_{j}=\frac{1}{2}\sum _{{\substack{k=1\\ k\neq j}}}^{N}\frac{\gamma _{j}\gamma _{k}}{a_{j}-a_{k}}-q_{j},$

$\tilde{\gamma}_{j}=\frac{\gamma _{j}}{2}\left(\frac{\gamma _{j}}{2}-1\right).$

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
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§31.14(ii) Kovacic’s Algorithm

Keywords:: Heun functions, Kovacic’s algorithm, spatio-temporal dynamics
Referenced by:: §31.8
Permalink:: http://dlmf.nist.gov/31.14.ii

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).

§31.14 General Fuchsian Equation

Contents

§31.14(i) Definitions

¶ Normal Form

§31.14(ii) Kovacic’s Algorithm