exponent parameters
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11—20 of 20 matching pages
11: Bibliography D
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Uniform asymptotic expansions for prolate spheroidal functions with large parameters.
SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
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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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Conical functions with one or both parameters large.
Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
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Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter
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Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.
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Uniform asymptotic expansions for oblate spheroidal functions II: Negative separation parameter
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Proc. Roy. Soc. Edinburgh Sect. A 125 (4), pp. 719–737.
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12: 15.11 Riemann’s Differential Equation
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15.11.2
►Here , , are the exponent pairs at the points , , , respectively.
Cases in which there are fewer than three singularities are included automatically by allowing the choice for exponent pairs.
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15.11.3
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15.11.4
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13: 36.10 Differential Equations
14: 15.10 Hypergeometric Differential Equation
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►It has regular singularities at , with corresponding exponent pairs , , , respectively.
When none of the exponent pairs differ by an integer, that is, when none of , , is an integer, we have the following pairs , of fundamental solutions.
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15.10.3
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15.10.5
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►(c) If the parameter
in the differential equation equals , then fundamental solutions in the neighborhood of are given by times those in (a) and (b), with and replaced throughout by and , respectively.
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15: 36.5 Stokes Sets
16: 36.6 Scaling Relations
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►For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).
17: 2.5 Mellin Transform Methods
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►where denotes the Bessel function (§10.2(ii)), and is a large positive parameter.
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►Then as in (2.5.6) and (2.5.7), with , we obtain
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§2.5(iii) Laplace Transforms with Small Parameters
… ►If in (2.5.18) and in (2.5.20), and if none of the exponents in (2.5.18) are positive integers, then the expansion (2.5.43) gives the following useful result: … ►For examples in which the integral defining the Mellin transform does not exist for any value of , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).18: 36.8 Convergent Series Expansions
19: 28.34 Methods of Computation
20: Bibliography M
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On one-parameter families of Painlevé III.
Stud. Appl. Math. 101 (3), pp. 321–341.
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Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions.
Comput. Phys. Comm. 178 (7), pp. 535–551.
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The characteristic numbers of the Mathieu equation with purely imaginary parameter.
Phil. Mag. Series 7 8 (53), pp. 834–840.
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Tables of the functions of the parabolic cylinder for negative integer parameters.
Zastos. Mat. 13, pp. 261–273.
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