infinite series

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21: 17.4 Basic Hypergeometric Functions

… ►The infinite series converges for all $z$ when $s>r$, and for when $s=r$. … ►The infinite series converge when $s\ge r$ provided that and also, in the case $s=r$, . …

22: 4.13 Lambert $W$-Function

… ►

4.13.10 $$W_k\left(z\right)\sim {\xi }_k-\ln {\xi }_k+\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{{\xi }_k^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln {\xi }_k\right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2).

Referenced by:

§4.13, §4.13, §4.45(iii), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infiniteseries.

See also:

Annotations for §4.13 and Ch.4

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4.13.11 $$W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim -\eta -\ln \eta +\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{\eta }^n}\sum _{m=1}^n\left[\genfrac{}{}{0.0pt}{}{n}{n-m+1}\right]\frac{{\left(-\ln \eta \right)}^m}{m!},$$

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$: Stirling cycle number of the first kind, $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$

Notes:

See Corless et al. (1996, p. 350).

Referenced by:

§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E11

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

Originally we gave only the first three terms of the infiniteseries.

See also:

Annotations for §4.13 and Ch.4

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23: 19.5 Maclaurin and Related Expansions

… ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product …

24: 20.5 Infinite Products and Related Results

… ►With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the $z$-plane. …

25: 8.17 Incomplete Beta Functions

… ►For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62). …

26: 13.9 Zeros

… ►Inequalities for ${\varphi }_r$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $M(a,b,x)$ are given in Ahmed and Muldoon (1980). …

27: Bibliography R

… ►

Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

External Links:

ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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28: 2.1 Definitions and Elementary Properties

… ►In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. …

29: 19.36 Methods of Computation

… ►If the iteration of (19.36.6) and (19.36.12) is stopped when ($M$ and $T$ being approximated by $a_s$ and $t_s$, and the infinite series being truncated), then the relative error in $R_F$ and $R_G$ is less than $r$ if we neglect terms of order $r^2$. …

30: 17.2 Calculus

… ►When $n\to \mathrm{\infty }$ in (17.2.35), and when $m\to \mathrm{\infty }$ in (17.2.38), the results become convergent infinite series and infinite products (see (17.5.1) and (17.5.4)). …