# solutions analytic at three singularities

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##### 1: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 SolutionsAnalyticatThreeSingularities: Heun Polynomials
is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. …
##### 2: 31.1 Special Notation
Sometimes the parameters are suppressed.
##### 3: 28.2 Definitions and Basic Properties
This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\infty$. …
###### §28.2(ii) Basic Solutions$w_{\mbox{\rm\tiny I}}$, $w_{\mbox{\rm\tiny II}}$
Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. Furthermore, a solution $w$ with given initial constant values of $w$ and $w^{\prime}$ at a point $z_{0}$ is an entire function of the three variables $z$, $a$, and $q$. The following three transformations …
##### 4: 2.8 Differential Equations with a Parameter
The form of the asymptotic expansion depends on the nature of the transition points in $\mathbf{D}$, that is, points at which $f(z)$ has a zero or singularity. … There are three main cases. …In Case II $f(z)$ has a simple zero at $z_{0}$ and $g(z)$ is analytic at $z_{0}$. In Case III $f(z)$ has a simple pole at $z_{0}$ and $(z-z_{0})^{2}g(z)$ is analytic at $z_{0}$. … The transformation is now specialized in such a way that: (a) $\xi$ and $z$ are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting $\psi(\xi)$ (or part of $\psi(\xi)$) has solutions that are functions of a single variable. …