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31 Heun FunctionsProperties

§31.14 General Fuchsian Equation


§31.14(i) Definitions

The general second-order Fuchsian equation with N+1 regular singularities at z=aj, j=1,2,,N, and at , is given by

31.14.1 d2wdz2+(j=1Nγjz-aj)dwdz+(j=1Nqjz-aj)w=0,

The exponents at the finite singularities aj are {0,1-γj} and those at are {α,β}, where

31.14.2 α+β+1 =j=1Nγj,
αβ =j=1Najqj.

The three sets of parameters comprise the singularity parameters aj, the exponent parameters α,β,γj, and the N-2 free accessory parameters qj. With a1=0 and a2=1 the total number of free parameters is 3N-3. Heun’s equation (31.2.1) corresponds to N=3.

Normal Form

31.14.3 w(z)=(j=1N(z-aj)-γj/2)W(z),
31.14.4 d2Wdz2=j=1N(γ~j(z-aj)2+q~jz-aj)W,
31.14.5 q~j =12k=1kjNγjγkaj-ak-qj,
γ~j =γj2(γj2-1).

§31.14(ii) Kovacic’s Algorithm

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).