from integral representations

SearchAdvancedHelp

(0.008 seconds)

11—20 of 79 matching pages

11: 8.19 Generalized Exponential Integral

… ►Integral representations of Mellin–Barnes type for $E_p\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1). …

12: 9.12 Scorer Functions

… ►where … ►

§9.12(vii) Integral Representations

… ►

Mellin–Barnes Type Integral

… ► … ►

Integrals

…

13: 22.20 Methods of Computation

… ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

14: 10.74 Methods of Computation

… ►

§10.74(iii) Integral Representations

►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). … ►The integral representation used is based on (10.32.8). ►For evaluation of $K_{\nu }\left(z\right)$ from (10.32.14) with $\nu =n$ and $z$ complex, see Mechel (1966). … ►

§10.74(vii) Integrals

…

15: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

… ►showing that the convergence of $c_n$ to 0 and of $a_n$ and $g_n$ to $M(a_0,g_0)$ is quadratic in each case. ►The AGM has the integral representations … ►

Descending Landen Transformation

… ►

Ascending Landen Transformation

…

16: 7.7 Integral Representations

§7.7 Integral Representations

►

§7.7(i) Error Functions and Dawson’s Integral

… ►

§7.7(ii) Auxiliary Functions

… ►

Mellin–Barnes Integrals

… ►For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).

17: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►Then the integral converges when provided that $z\ne 0$, or when $p=q+1$ provided that , and provides an integral representation of the left-hand side with these conditions. ►Secondly, suppose that $L$ is a contour from $-\mathrm{i}\mathrm{\infty }$ to $\mathrm{i}\mathrm{\infty }$. …In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when . … ►For further integral representations and integrals see Apelblat (1983, §16), Erdélyi et al. (1953a, §4.6), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), and Prudnikov et al. (1990, §§2.22, 4.2.4, and 4.3.1). …

18: 15.6 Integral Representations

§15.6 Integral Representations

►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: … ►Note that (15.6.8) can be rewritten as a fractional integral. … ►In (15.6.6) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$, and ${(-z)}^t$ has its principal value. ►In (15.6.7) the integration contour separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ and $\mathrm{\Gamma }\left(b+t\right)$ from those of $\mathrm{\Gamma }\left(c-a-b-t\right)$ and $\mathrm{\Gamma }\left(-t\right)$, and ${(1-z)}^t$ has its principal value. …

19: Bibliography V

… ►

Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 72, Kluwer Academic Publishers Group, Dordrecht.

ⓘ

Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1466-2, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

ⓘ

Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

…

20: Bibliography E

… ►

Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.

ⓘ

Notes:

Translated from the Russian by H. H. McFadden, Translation edited by Lev J. Leifman

External Links:

ISBN 0-8218-4512-8, MathReview, ZentralBlatt

Referenced by:

§15.17(iv).

Encodings:

BibTeX

…