# confluence of singularities

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## 3 matching pages

##### 1: 16.8 Differential Equations

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###### §16.8(iii) Confluence of Singularities

…##### 2: 31.12 Confluent Forms of Heun’s Equation

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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
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►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty}$.
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►This has irregular singularities at $z=0$ and $\mathrm{\infty}$, each of rank $1$.
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►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty}$ of rank $2$.
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►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty}$.
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##### 3: 2.4 Contour Integrals

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►The function $f(\alpha ,w)$ is analytic at $w={w}_{1}(\alpha )$ and $w={w}_{2}(\alpha )$ when $\alpha \ne \widehat{\alpha}$, and at the confluence of these points when $\alpha =\widehat{\alpha}$.
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►The problems sketched in §§2.3(v) and 2.4(v) involve only two of many possibilities for the coalescence of endpoints, saddle points, and singularities in integrals associated with the special functions.
…For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions.
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►For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998).
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