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31 Heun FunctionsProperties

§31.2 Differential Equations

  1. §31.2(i) Heun’s Equation
  2. §31.2(ii) Normal Form of Heun’s Equation
  3. §31.2(iii) Trigonometric Form
  4. §31.2(iv) Doubly-Periodic Forms
  5. §31.2(v) Heun’s Equation Automorphisms

§31.2(i) Heun’s Equation

31.2.1 d2wdz2+(γz+δz1+ϵza)dwdz+αβzqz(z1)(za)w=0,

This equation has regular singularities at 0,1,a,, with corresponding exponents {0,1γ}, {0,1δ}, {0,1ϵ}, {α,β}, respectively (§2.7(i)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, {}, can be transformed into (31.2.1).

The parameters play different roles: a is the singularity parameter; α,β,γ,δ,ϵ are exponent parameters; q is the accessory parameter. The total number of free parameters is six.

§31.2(ii) Normal Form of Heun’s Equation

31.2.2 w(z)=zγ/2(z1)δ/2(za)ϵ/2W(z),
31.2.3 d2Wdz2=(Az+Bz1+Cza+Dz2+E(z1)2+F(za)2)W,
31.2.4 A =γδ2γϵ2a+qa,
B =γδ2δϵ2(a1)qαβa1,
C =γϵ2a+δϵ2(a1)aαβqa(a1),
D =12γ(12γ1),
E =12δ(12δ1),
F =12ϵ(12ϵ1).

§31.2(iii) Trigonometric Form

31.2.5 z=sin2θ,
31.2.6 d2wdθ2+((2γ1)cotθ(2δ1)tanθϵsin(2θ)asin2θ)dwdθ+4αβsin2θqasin2θw=0.

§31.2(iv) Doubly-Periodic Forms

Jacobi’s Elliptic Form

With the notation of §22.2 let

31.2.7 a =k2,
z =sn2(ζ,k).

Then (suppressing the parameter k)

31.2.8 d2wdζ2+((2γ1)cnζdnζsnζ(2δ1)snζdnζcnζ(2ϵ1)k2snζcnζdnζ)dwdζ+4k2(αβsn2ζq)w=0.

Weierstrass’s Form

With the notation of §§19.2(ii) and 23.2 let

31.2.9 k2 =(e2e3)/(e1e3),
ζ =iK+ξ(e1e3)1/2,
e1 =(ω1),
e2 =(ω2),
e3 =(ω3),

where 2ω1 and 2ω3 with (ω3/ω1)>0 are generators of the lattice 𝕃 for (z|𝕃). Then

31.2.10 w(ξ)=((ξ)e3)(12γ)/4((ξ)e2)(12δ)/4((ξ)e1)(12ϵ)/4W(ξ),

where W(ξ) satisfies

31.2.11 d2W/dξ2+(H+b0(ξ)+b1(ξ+ω1)+b2(ξ+ω2)+b3(ξ+ω3))W=0,


31.2.12 b0 =4αβ(γ+δ+ϵ12)(γ+δ+ϵ32),
b1 =(ϵ12)(ϵ32),
b2 =(δ12)(δ32),
b3 =(γ12)(γ32),
H =e1(γ+δ1)2+e2(γ+ϵ1)2+e3(δ+ϵ1)24αβe34q(e2e3).

§31.2(v) Heun’s Equation Automorphisms

F-Homotopic Transformations

w(z)=z1γw1(z) satisfies (31.2.1) if w1 is a solution of (31.2.1) with transformed parameters q1=q+(aδ+ϵ)(1γ); α1=α+1γ, β1=β+1γ, γ1=2γ. Next, w(z)=(z1)1δw2(z) satisfies (31.2.1) if w2 is a solution of (31.2.1) with transformed parameters q2=q+aγ(1δ); α2=α+1δ, β2=β+1δ, δ2=2δ. Lastly, w(z)=(za)1ϵw3(z) satisfies (31.2.1) if w3 is a solution of (31.2.1) with transformed parameters q3=q+γ(1ϵ); α3=α+1ϵ, β3=β+1ϵ, ϵ3=2ϵ. By composing these three steps, there result 23=8 possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).

Homographic Transformations

There are 4!=24 homographies z~(z)=(Az+B)/(Cz+D) that take 0,1,a, to some permutation of 0,1,a,, where a may differ from a. If z~=z~(z) is one of the 3!=6 homographies that map to , then w(z)=w~(z~) satisfies (31.2.1) if w~(z~) is a solution of (31.2.1) with z replaced by z~ and appropriately transformed parameters. For example, if z~=z/a, then the parameters are a~=1/a, q~=q/a; δ~=ϵ, ϵ~=δ. If z~=z~(z) is one of the 4!3!=18 homographies that do not map to , then an appropriate prefactor must be included on the right-hand side. For example, w(z)=(1z)αw~(z/(z1)), which arises from z~=z/(z1), satisfies (31.2.1) if w~(z~) is a solution of (31.2.1) with z replaced by z~ and transformed parameters a~=a/(a1), q~=(qaαγ)/(a1); β~=α+1δ, δ~=α+1β.

Composite Transformations

There are 824=192 automorphisms of equation (31.2.1) by compositions of F-homotopic and homographic transformations. Each is a substitution of dependent and/or independent variables that preserves the form of (31.2.1). Except for the identity automorphism, each alters the parameters.