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§31.2 Differential Equations


§31.2(i) Heun’s Equation

31.2.1 d2wdz2+(γz+δz-1+ϵz-a)dwdz+αβz-qz(z-1)(z-a)w=0,

This equation has regular singularities at 0,1,a,, with corresponding exponents {0,1-γ}, {0,1-δ}, {0,1-ϵ}, {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)=z-γ/2(z-1)-δ/2(z-a)-ϵ/2W(z),
31.2.3 d2Wdz2=(Az+Bz-1+Cz-a+Dz2+E(z-1)2+F(z-a)2)W,
31.2.4 A =-γδ2-γϵ2a+qa,
B =γδ2-δϵ2(a-1)-q-αβa-1,
C =γϵ2a+δϵ2(a-1)-aαβ-qa(a-1),
D =12γ(12γ-1),
E =12δ(12δ-1),
F =12ϵ(12ϵ-1).

§31.2(iii) Trigonometric Form

§31.2(iv) Doubly-Periodic Forms

Jacobi’s Elliptic Form

With the notation of §22.2 let

31.2.7 a =k-2,
z =sn2(ζ,k).

Then (suppressing the parameter k)

31.2.8 d2wdζ2+((2γ-1)cnζdnζsnζ-(2δ-1)snζdnζcnζ-(2ϵ-1)k2snζcnζdnζ)dwdζ+4k2(αβsn2ζ-q)w=0.

Weierstrass’s Form

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

31.2.9 k2 =(e2-e3)/(e1-e3),
ζ =iK+ξ(e1-e3)1/2,
e1 =(ω1),
e2 =(ω2),
e3 =(ω3),
e1+e2+e3 =0,

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)(1-2γ)/4((ξ)-e2)(1-2δ)/4((ξ)-e1)(1-2ϵ)/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+ϵ-12)(γ+δ+ϵ-32),
b1 =-(ϵ-12)(ϵ-32),
b2 =-(δ-12)(δ-32),
b3 =-(γ-12)(γ-32),
H =e1(γ+δ-1)2+e2(γ+ϵ-1)2+e3(δ+ϵ-1)2-4αβe3-4q(e2-e3).

§31.2(v) Heun’s Equation Automorphisms

F-Homotopic Transformations

w(z)=z1-γw1(z) satisfies (31.2.1) if w1 is a solution of (31.2.1) with transformed parameters q1=q+(aδ+ϵ)(1-γ); α1=α+1-γ, β1=β+1-γ, γ1=2-γ. Next, w(z)=(z-1)1-δw2(z) satisfies (31.2.1) if w2 is a solution of (31.2.1) with transformed parameters q2=q+aγ(1-δ); α2=α+1-δ, β2=β+1-δ, δ2=2-δ. Lastly, w(z)=(z-a)1-ϵw3(z) satisfies (31.2.1) if w3 is a solution of (31.2.1) with transformed parameters q3=q+γ(1-ϵ); α3=α+1-ϵ, β3=β+1-ϵ, ϵ3=2-ϵ. 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)=(1-z)-αw~(z/(z-1)), which arises from z~=z/(z-1), satisfies (31.2.1) if w~(z~) is a solution of (31.2.1) with z replaced by z~ and transformed parameters a~=a/(a-1), q~=-(q-aαγ)/(a-1); β~=α+1-δ, δ~=α+1-β.

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.