Heun function

Remarks

Heun's equation is a Fuchsian equation with regular singular points $0, 1, a, \infty$. Every Fuchsian equation with four regular singular points takes this form after moving the singular points there and making one exponent $0$ at each finite point. The exponents are $\set{0, 1 - \gamma}$ at $0$, $\set{0, 1 - \delta}$ at $1$, $\set{0, 1 - \epsilon}$ at $a$ and $\set{\alpha, \beta}$ at $\infty$. The relation $\epsilon = \alpha + \beta + 1 - \gamma - \delta$ is Fuchs' relation: the sum of all exponents equals $2$. The number $q$ is the accessory parameter; it is not determined by the exponents. The power series of $Hl$ at $0$ converges in the disc $\abs{z} < \min(1, \abs{a})$. Example: for $q = 0$ and $\alpha = 0$, the constant $w = 1$ is a solution, since $w' = w'' = 0$ and $\alpha\beta z - q = 0$; so $Hl(a, 0; 0, \beta, \gamma, \delta; z) = 1$. Non-example: for $q = 1$ and $\alpha = 0$, the constant $1$ is not a solution, since the equation leaves the term $\frac{-1}{z (z - 1)(z - a)} \ne 0$. A Heun function in the narrow sense is a local solution that is analytic at two singular points at once, for instance at $0$ and $1$. This happens only for special values of $q$.