integral transforms in terms of
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11: 2.6 Distributional Methods
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►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term.
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►The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals.
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§2.6(ii) Stieltjes Transform
… ►Corresponding results for the generalized Stieltjes transform … ►The replacement of by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form …12: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
… ►with . … ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms and to have a common strip of analyticity. … ►Next from Table 2.5.1 we observe that the integrals for the transform pair and are absolutely convergent in the domain specified in Table 2.5.2, and these domains are nonempty as a consequence of (2.5.19) and (2.5.20). … ►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).13: 2.3 Integrals of a Real Variable
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►Assume that the Laplace transform
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►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion:
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►In the integral
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►The integral (2.3.24) transforms into
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§2.3(vi) Asymptotics of Mellin Transforms
…14: 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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The Radon Transform and Some of Its Applications.
A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.
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Integral transforms and their applications.
Third edition, CRC Press, Boca Raton, FL.
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Infinite integrals in the theory of Bessel functions.
Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.
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Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities.
Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.
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15: Bibliography W
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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations.
Computers in Physics 10 (5), pp. 496–503.
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A class of integral transforms.
Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.
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Sequence Transformations and their Applications.
Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.
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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials.
Anal. Appl. (Singap.) 1 (2), pp. 213–241.
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Quadrature formulas for oscillatory integral transforms.
Numer. Math. 39 (3), pp. 351–360.
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16: Bibliography L
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Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions.
J. Electromagn. Waves Appl. 12 (6), pp. 709–711.
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Approximation of orthogonal polynomials in terms of Hermite polynomials.
Methods Appl. Anal. 6 (2), pp. 131–146.
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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
Stud. Appl. Math. 103 (3), pp. 241–258.
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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions.
Proc. Amer. Math. Soc. 108 (2), pp. 387–389.
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Integral representation of the Hankel function in terms of parabolic cylinder functions.
Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.
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17: 1.8 Fourier Series
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►Here is related to and
in (1.8.1), (1.8.2) by , for and .
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►(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in
, provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large .
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►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval.
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►If a function is periodic, with period , then the series obtained by differentiating the Fourier series for
term by term converges at every point to .
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►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to .
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18: 23.6 Relations to Other Functions
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►In (23.6.27)–(23.6.29) the modulus is given and , are the corresponding complete elliptic integrals (§19.2(ii)).
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►Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.
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►For representations of general elliptic functions (§23.2(iii)) in terms of and see Lawden (1989, §§8.9, 8.10), and for expansions in terms of see Lawden (1989, §8.11).
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§23.6(iv) Elliptic Integrals
… ►For relations to symmetric elliptic integrals see §19.25(vi). …19: Bibliography O
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Tables of Fourier Transforms and Fourier Transforms of Distributions.
Springer-Verlag, Berlin.
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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles.
Methods Appl. Anal. 7 (4), pp. 727–745.
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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions.
Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms.
SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
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The Generalized Exponential Integral.
In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.),
International Series of Numerical Mathematics, Vol. 119, pp. 497–510.
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20: 19.25 Relations to Other Functions
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►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14).
…Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions).
►The three changes of parameter of
in §19.7(iii) are unified in (19.21.12) by way of (19.25.14).
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