Heun's equation

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

$\frac {d^2w}{dz^2} + \left[\frac{\gamma}{z}+ \frac{\delta}{z-1} + \frac{\epsilon}{z-d} \right] \frac {dw}{dz} + \frac {\alpha \beta z -q} {z(z-1)(z-d)} w = 0.$

(Note that $ε = α + β - γ - δ + 1$ is needed to ensure regularity of the point at ∞.) Every second-order linear ODE in the complex plane (or on the Riemann sphere, to be more accurate) with four regular singular points can be transformed into this equation. It has four regular singular points: 0,1,d and ∞.

The equation is named after Karl L. W. M. Heun. ("Heun" rhymes with "loin.")