# solutions analytic at three singularities

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## 4 matching pages

##### 1: 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 $0,1,a$. …##### 2: 31.1 Special Notation

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►Sometimes the parameters are suppressed.

##### 3: 28.2 Definitions and Basic Properties

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►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at
$\mathrm{\infty}$.
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###### §28.2(ii) Basic Solutions ${w}_{\text{I}}$, ${w}_{\text{II}}$

►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

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►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. …