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confluence of singularities

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1: 16.8 Differential Equations
§16.8(iii) Confluence of Singularities
2: 31.12 Confluent Forms of Heun’s Equation
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. … This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . … This has irregular singularities at z = 0 and , each of rank 1 . … This has a regular singularity at z = 0 , and an irregular singularity at of rank 2 . … This has one singularity, an irregular singularity of rank 3 at z = . …
3: 2.4 Contour Integrals
The function f ( α , w ) is analytic at w = w 1 ( α ) and w = w 2 ( α ) when α α ^ , and at the confluence of these points when α = α ^ . … 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. … For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). …