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21: 35.2 Laplace Transform
§35.2 Laplace Transform
βΊDefinition
… βΊInversion Formula
… βΊConvolution Theorem
βΊIf is the Laplace transform of , , then is the Laplace transform of the convolution , where …22: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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23: Bibliography G
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Problem 72-21, Laplace transforms of Airy functions.
SIAM Rev. 15 (4), pp. 796–798.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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Fourier transforms related to a root system of rank 1.
Transform. Groups 12 (1), pp. 77–116.
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24: Bibliography P
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A Kummer-type transformation for a hypergeometric function.
J. Comput. Appl. Math. 173 (2), pp. 379–382.
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Bounds for ratios of modified Bessel functions.
Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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Fourier Series and Integral Transforms.
Cambridge University Press, Cambridge.
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Integrals and Series: Direct Laplace Transforms, Vol. 4.
Gordon and Breach Science Publishers, New York.
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25: 15.17 Mathematical Applications
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βΊThe logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations.
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βΊHarmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform.
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βΊQuadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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βΊBy considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group.
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26: 19.15 Advantages of Symmetry
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βΊSymmetry in of , , and replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17).
Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9).
(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral.
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27: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(vi) Direct Numerical Transformations
βΊThe transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … βΊ … βΊFor example, using double precision is found to agree with (2.11.31) to 13D. βΊHowever, direct numerical transformations need to be used with care. …28: 32.8 Rational Solutions
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– possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants.
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βΊRational solutions of exist for and are generated using the seed solution and the Bäcklund transformations (32.7.1) and (32.7.2).
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32.8.3
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32.8.4
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29: Bibliography D
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Integral Transforms and their Applications.
2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
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Integral transforms and their applications.
Third edition, CRC Press, Boca Raton, FL.
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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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30: 36 Integrals with Coalescing Saddles
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