confluence of singularities
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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 and , and an irregular singularity of rank 1 at .
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►This has irregular singularities at and , each of rank .
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►This has a regular singularity at , and an irregular singularity at of rank .
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►This has one singularity, an irregular singularity of rank at .
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3: 2.4 Contour Integrals
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►The function is analytic at and when , and at the confluence of these points when .
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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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