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coalescing peak and endpoint


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1: 2.3 Integrals of a Real Variable
§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method
2: 2.4 Contour Integrals
The problem of obtaining an asymptotic approximation to I ( α , z ) that is uniform with respect to α in a region containing α ^ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). … 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 endpoint see Leubner and Ritsch (1986). …
3: 7.20 Mathematical Applications
For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). …
4: Bibliography L
  • C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.