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11: 7.20 Mathematical Applications
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►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).
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12: 36.13 Kelvin’s Ship-Wave Pattern
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►These coalesce when
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►The disturbance can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that are real for and complex for .
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13: 36.4 Bifurcation Sets
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►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.1) and
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►This is the codimension-one surface in space where critical points coalesce, satisfying (36.4.2) and
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14: 8.13 Zeros
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►As increases the positive zeros coalesce to form a double zero at ().
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15: 32.2 Differential Equations
16: Bibliography D
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function.
Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
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17: Bibliography N
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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole.
Ph.D. Thesis, University of Maryland, College Park, MD.
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18: 2.8 Differential Equations with a Parameter
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§2.8(vi) Coalescing Transition Points
… ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. ►For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …19: 29.3 Definitions and Basic Properties
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►The eigenvalues coalesce according to
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20: 2.3 Integrals of a Real Variable
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