integral transforms in terms of

SearchAdvancedHelp

(0.012 seconds)

11—20 of 48 matching pages

11: 2.6 Distributional Methods

… ►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. … ►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. … ►

§2.6(ii) Stieltjes Transform

… ►Corresponding results for the generalized Stieltjes transform … ►The replacement of $f(t)$ 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 $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ to have a common strip of analyticity. … ►Next from Table 2.5.1 we observe that the integrals for the transform pair $ℳf_j\left(1-z\right)$ and $ℳh_k\left(z\right)$ are absolutely convergent in the domain $D_{jk}$ 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 $ℳh\left(z\right)$ does not exist for any value of $z$, see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).

13: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►In the integral … ►The integral (2.3.24) transforms into … ►

§2.3(vi) Asymptotics of Mellin Transforms

…

14: Bibliography D

… ►

Integral Transforms and their Applications. 2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-96080-5, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

… ►

The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

ⓘ

Notes:

Reprinted by Dover in 2007.

External Links:

ISBN 0-471-89804-X, MathReview (W. R. Madych), ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

►

Integral transforms and their applications. Third edition, CRC Press, Boca Raton, FL.

ⓘ

External Links:

ISBN 978-1-4822-2357-6, MathReview Entry, ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

… ►

Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview (Alexander Tovbis), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

…

15: Bibliography W

… ►

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.

ⓘ

External Links:

Document

Referenced by:

§2.11(vi), §2.11(vi).

Encodings:

BibTeX

… ►

A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

ⓘ

External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

Encodings:

BibTeX

… ►

Sequence Transformations and their Applications. Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.

ⓘ

External Links:

ISBN 0-12-757940-0, MathReview (Jan Zítko), ZentralBlatt

Referenced by:

§3.9(vi).

Encodings:

BibTeX

… ►

Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

… ►

Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

ⓘ

External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

BibTeX

…

16: Bibliography L

… ►

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.

ⓘ

External Links:

ISSN 0920-5071, Document, MathReview

Referenced by:

§30.14(v).

Encodings:

BibTeX

… ►

Approximation of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6 (2), pp. 131–146.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§18.15(vi), §18.16(vi).

Encodings:

BibTeX

… ►

Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

ⓘ

External Links:

ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

Encodings:

BibTeX

… ►

Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

ⓘ

External Links:

ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

Encodings:

BibTeX

… ►

Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

ⓘ

External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

BibTeX

…

17: 1.8 Fourier Series

… ►Here $c_n$ is related to $a_n$ and $b_n$ in (1.8.1), (1.8.2) by $c_n=\frac{1}{2}(a_n-\mathrm{i}{b}_n)$, $c_{-n}=\frac{1}{2}(a_n+\mathrm{i}{b}_n)$ for $n>0$ and $c_0=\frac{1}{2}{a}_0$. … ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. … ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. …

18: 23.6 Relations to Other Functions

… ►In (23.6.27)–(23.6.29) the modulus $k$ is given and $K=K\left(k\right)$, $K^{\prime }=K\left(k^{\prime }\right)$ are the corresponding complete elliptic integrals (§19.2(ii)). … ►Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3. … ►For representations of general elliptic functions (§23.2(iii)) in terms of $\sigma \left(z\right)$ and $\mathrm{\wp }\left(z\right)$ see Lawden (1989, §§8.9, 8.10), and for expansions in terms of $\zeta \left(z\right)$ see Lawden (1989, §8.11). ►

§23.6(iv) Elliptic Integrals

… ►For relations to symmetric elliptic integrals see §19.25(vi). …

19: Bibliography O

… ►

Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

ⓘ

External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

Encodings:

BibTeX

… ►

On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

Encodings:

BibTeX

… ►

Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§2.11(iii), §8.19(viii).

Encodings:

BibTeX

…

20: 19.25 Relations to Other Functions

… ►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 $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). ►

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ►For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9). …