 Heun equation

The general Heun equation is the second-order linear ordinary differential equationIt is a canonical Fuchsian equation with four singular locations $0,1,a,\infty$, generically all regular singular points. Special parameter choices can make some singularities apparent or removable. The local exponent pairs are $(0,1-\gamma )$, $(0,1-\delta )$, $(0,1-ϵ)$ and $(\alpha ,\beta )$, respectively.^[ref]

The parameter $a$ locates the fourth singularity, while $q$ is an accessory parameter not fixed by those local exponent pairs. For instance, setting $\alpha =q=0$ makes the constant function $w=1$ a solution, as direct substitution shows. Coalescing singularities gives confluent Heun equations with different singularity structures; these are not the same four-distinct-point equation. Heun functions supply named solutions for boundary-value problems beyond the three-singularity hypergeometric case.