Borel transform theory

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1: 1.14 Integral Transforms

§1.14 Integral Transforms

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§1.14(i) Fourier Transform

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§1.14(iii) Laplace Transform

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Fourier Transform

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Laplace Transform

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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

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Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.

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External Links:

ZentralBlatt

Referenced by:

§16.24(i).

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Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:

§36.14(i), §9.16.

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Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

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FullText

Referenced by:

2nd item.

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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

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Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

Encodings:

BibTeX

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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:

ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

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BibTeX

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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

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§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(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

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Borel Summability

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1.15.36 $$h(x,y)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-y\left|t\right|}{\mathrm{e}}^{-\mathrm{i}xt}F(t)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ , $h(x,y)$: function and $\left|x\right|$: absolute value of $x$

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http://dlmf.nist.gov/1.15.E36

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Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

►where $F(t)$ is the Fourier transform of $f(x)$ (§1.14(i)). … ►Moreover, ${lim}_{y\to 0+}\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)). … ►

1.15.44 $${\sigma }_R(\theta )=\frac{1}{\sqrt{2\pi }}{\int }_{-R}^R\left(1-\frac{\left|t\right|}{R}\right){\mathrm{e}}^{-\mathrm{i}\theta t}F(t)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ , ${\sigma }_R(\theta )$: function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E44

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See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

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8: 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 transform (§1.14(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. …

9: 19.15 Advantages of Symmetry

… ►The function $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ (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)). …

10: Mourad E. H. Ismail

… ►Ismail has published numerous papers on special functions, orthogonal polynomials, approximation theory, combinatorics, asymptotics, and related topics. … ► Suslov), Kluwer Academic Publishers, 2001; Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (with F. … Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special Functions: A volume dedicated to Mizan Rahman (with E. … ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …