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11: 2.5 Mellin Transform Methods

… ►The Mellin transform of $f(t)$ is defined by … ►We now apply (2.5.5) with , and then translate the integration contour to the right. … ►First, we introduce the truncated functions $f_1(t)$ and $f_2(t)$ defined by …With these definitions and the conditions (2.5.17)–(2.5.20) the Mellin transforms converge absolutely and define analytic functions in the half-planes shown in Table 2.5.1. … ►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).

12: Bibliography F

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

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Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

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External Links:

ISSN 1058-6458, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.16.13), §25.16(ii).

Encodings:

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

Encodings:

BibTeX

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13: 12.14 The Function $W(a,x)$

… ►the branch of $\mathrm{ph}$ being zero when $a=0$ and defined by continuity elsewhere. … ►

§12.14(vi) Integral Representations

►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►Here ${𝒜}_s(t)$ is as in §12.10(ii), $\sigma$ is defined by … ►where $k$ is defined in (12.14.5), and $\tilde{F}(a,x)$ ($>$0), $\tilde{\theta }(a,x)$, $\tilde{G}(a,x)$ ($>$0), and $\tilde{\psi }(a,x)$ are real. …

14: 16.2 Definition and Analytic Properties

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … ►See §16.5 for the definition of ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. …