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solutions analytic at three singularities

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1: 31.1 Special Notation
Sometimes the parameters are suppressed.
2: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: 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 . …
3: 28.2 Definitions and Basic Properties
This equation has regular singularities at 0 and 1, both with exponents 0 and 1 2 , and an irregular singular point at . …
§28.2(ii) Basic Solutions w I , w 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 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 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) ξ and z are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting ψ ( ξ ) (or part of ψ ( ξ ) ) has solutions that are functions of a single variable. …