休恩函数

Heun函数指HeunB、HeunC、HeunD、HeunG、HeunT等五个函数

HeunG 函数
HeunG函数**是下列GeneralHeun方程的解

:\frac {d^2w}{dz^2} +
\left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right]
\frac {dw}{dz}

• \frac {\alpha \beta z -q} {z(z-1)(z-a)} w = 0.

奇点
HeunG(a, b, \alpha, \beta, \gamma, \delta, z)函数有6个参数，有四个奇点

展开式
HeunG(a, b, \alpha, \beta, \gamma, \delta, z) =bz/(\gammaa)-(1/2)(-b-ba\delta-b\gammaa+b\delta-b\alpha-b\beta-b^2+\alpha\beta\gammaa)z^2/(\gammaa^2(\gamma+1))+O(z^3)
关系式
:HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(1-\delta)HeunG(a, q-(-1+\delta)\gamma*a, \beta-\delta+1, \alpha-\delta+1, \gamma, 2-\delta, z)
: [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z/a)^(-\alpha-\beta+\gamma+\delta)HeunG(a, q-\gamma(\alpha+\beta-\gamma-\delta), -\beta+\gamma+\delta, -\alpha+\gamma+\delta, \gamma, \delta, z), And(a <> 0)]
:[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(1-\delta)(1-z/a)^(-\alpha-\beta+\gamma+\delta)HeunG(a, q-\gamma((-1+\delta)a+\alpha+\beta-\gamma-\delta), -\beta+\gamma+1, -\alpha+\gamma+1, \gamma, 2-\delta, z), And(a <> 0)]
: [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = HeunG(1/a, q/a, \alpha, \beta, \gamma, \alpha+\beta-\gamma-\delta+1, z/a), And(a <> 0)]
: [HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = (1-z)^(-\alpha)HeunG(a/(-1+a), (-q+\gamma\alpha*a)/(-1+a), \alpha, \alpha-\delta+1, \gamma, \alpha-\beta+1, z/(z-1)), And(a <> 1, z <> 1)]
:
超几何函数关系
:[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = hypergeom([\alpha, \beta], [\alpha+\beta-\delta+1], z), And(a = 0, q = 0)]
:[HeunG(a, q, \alpha, \beta, \gamma, \delta, z) = hypergeom([\alpha, \beta], [\gamma], z)(And(a = 1, q = \alpha\beta), And(q = \alpha\beta*a, \delta = \alpha+\beta-\gamma+1))]
HeunB函数
HeunB(a,b,c,d,z)是下列双合流方程的解
y_zz-\frac{(\alpha+2z+z^2\alpha-2z^3)}{(z+1)^2/(z-1)^2}y_z+\frac{(\delta+(2\alpha+\gamma)z+\betaz^2)}{(z-1)^3/(z+1)^3y(z)};

HeunC函数
HeunC(a,b,c,d,z)是下列微分方程的解
y_zz-\frac{(-z^2a+(-2-b-c+a)z+b+1)y_z)}{(z(z-1))}-\frac{(1/2)(((-b-c-2)a-2d)z+(b+1)a+(-c-1)b-c-2e)y}{(z*(z-1))}
关系式
:[HeunC(a, b, c, d, e, z) = (1-z)^(-c)*HeunC(a, b, -c, d, e, z), And(b::(Not(integer)), abs(z)
:[HeunC(a, b, c, d, e, z) = exp(-za)HeunC(-a, b, c, d, e, z), And(b::(Not(integer)), abs(z)

HeunD函数
HeunD(a,b,c,d,z)函数是下列微分方程的解
y_zz-\frac{(-2z^5+4z^3+z^4a-2z-a)y_z)}{((z-1)^3(z+1)^3)}-\frac{(-z^2b+(-c-2a)z-d)y}{((z-1)^3*(z+1)^3)}
关系式
[HeunD(a, b, c, d, z) = HeunD(-a, -d, -c, -b, 1/z)]

HeunT函数
HeunT(a,b,c,z)函数是下列微分方程的解

y_zz-(3z^2+c)y_z-((3-b)z-a)y

关系式
[HeunT(a, b, c, z) = HeunT(ja, b, j^2c, j*z), And(j^3 = 1)]
[HeunT(a, b, c, z) = HeunT(a, -b, c, -z)*exp(z^3), And(c = 0)]

参考文献
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