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Borel transform theory


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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 2.11 Remainder Terms; Stokes Phenomenon
For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … The first of these two references also provides an introduction to the powerful Borel transform theory. … However, direct numerical transformations need to be used with care. …
3: Bibliography B
  • M. N. Barber and B. W. Ninham (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.
  • M. V. Berry (1969) Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.
  • D. Bleichenbacher (1996) Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.
  • W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
  • J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.
  • 4: 12.16 Mathematical Applications
    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). … PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).
    5: 27.17 Other Applications
    §27.17 Other Applications
    Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. … There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.
    6: 14.31 Other Applications
    §14.31(ii) Conical Functions
    These functions are also used in the Mehler–Fock integral transform14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
    7: 1.15 Summability Methods
    Borel Summability
    1.15.36 h ( x , y ) = 1 2 π - e - y | t | e - i x t F ( t ) d t ,
    where F ( t ) is the Fourier transform of f ( x ) 1.14(i)). … Moreover, lim y 0 + Φ ( x + i y ) is the Hilbert transform of f ( x ) 1.14(v)). …
    1.15.44 σ R ( θ ) = 1 2 π - R R ( 1 - | t | R ) e - i θ t F ( t ) d t ,
    8: 18.38 Mathematical Applications
    Approximation Theory
    Integrable Systems
    Complex Function Theory
    Random Matrix Theory
    Coding Theory
    9: 15.17 Mathematical Applications
    The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
    §15.17(v) Monodromy Groups
    By 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. …
    10: 19.15 Advantages of Symmetry
    The function R - a ( b 1 , b 2 , , b n ; z 1 , z 2 , , z n ) (Carlson (1963)) reveals the full permutation symmetry that is partially hidden in F D , and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. … 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. … These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …