kernel%20equations
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11: 12.16 Mathematical Applications
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►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi).
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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12: 13.27 Mathematical Applications
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►The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions.
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►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).
13: 10.63 Recurrence Relations and Derivatives
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►Equations (10.63.6) and (10.63.7) also hold when the symbols and in (10.63.5) are replaced throughout by and , respectively.
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§10.63(i) , , ,
… ►14: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…15: 28.10 Integral Equations
§28.10 Integral Equations
►§28.10(i) Equations with Elementary Kernels
… ►§28.10(ii) Equations with Bessel-Function Kernels
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28.10.10
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§28.10(iii) Further Equations
…16: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials.
Comm. Math. Phys. 398 (3), pp. 1291–1348.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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17: 10.75 Tables
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Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Abramowitz and Stegun (1964, Chapter 9) tabulates , , , , , , 9–10D; , , , , 9D; modulus and phase functions , , , , , , 6D; , , , , , , 5D.
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.
Zhang and Jin (1996, p. 323) tabulates the first real zeros of , , , , , , , , 8D.
18: Bibliography W
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Solutions of the fifth Painlevé equation. I.
Hokkaido Math. J. 24 (2), pp. 231–267.
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On a generalization of the functions x, x, x, x.
Quart. J. Pure Appl. Math. 42, pp. 316–342.
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On the central connection problem for the double confluent Heun equation.
Math. Nachr. 195, pp. 267–276.
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19: 10.62 Graphs
20: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels.
SIAM J. Math. Anal. 11 (5), pp. 828–841.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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