Heun Function - Heun's Equation

Heun's Equation

Heun's equation is a second-order linear ordinary differential equation (ODE) of the form

$\frac {d^2w}{dz^2} + \left \frac {dw}{dz} + \frac {\alpha \beta z -q} {z(z-1)(z-a)} w = 0.$

The condition is needed to ensure regularity of the point at ∞.

The complex number q is called the accessory parameter. Heun's equation has four regular singular points: 0, 1, a and ∞ with exponents (0, 1 − γ), (0, 1 − δ), (0, 1 − ϵ), and (α, β). Every second-order linear ODE on the extended complex plane with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation, can be transformed into this equation by a change of variable.

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