DLMF: Untitled Document

… ►

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►

b_{0}+\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

is equivalent to

b^{\prime}_{0}+\displaystyle{\cfrac{a^{\prime}_{1}}{b^{\prime}_{1}+\cfrac{a^{% \prime}_{2}}{b^{\prime}_{2}+\cdots}}}

if there is a sequence

\{d_{n}\}^{\infty}_{n=0}

,

d_{0}=1

,

d_{n}\neq 0

, such that … ►The even part of

C

exists iff

b_{2k}\not=0

,

k=1,2,\dots

, and up to equivalence is given by … ►The continued fraction

\displaystyle{\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}}

converges when … ►Let the elements of the continued fraction

\displaystyle{\cfrac{1}{b_{1}+\cfrac{1}{b_{2}+\cdots}}}

satisfy …

… ►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. For example, suppose a lengthy calculation involves many 10-digit integers. …Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

… ►Gergő Nemes (b. … ►He obtained a B. … in mathematics (with distinction) and a M. …in mathematics (with honours) from Loránd Eötvös University, Budapest, Hungary and a Ph. … ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

… ►It has regular singularities at

z=0,1,\infty

, with corresponding exponent pairs

\{0,1-c\}

,

\{0,c-a-b\}

,

\{a,b\}

, respectively. When none of the exponent pairs differ by an integer, that is, when none of

c

,

c-a-b

,

a-b

is an integer, we have the following pairs

f_{1}(z)

,

f_{2}(z)

of fundamental solutions. … ►Moreover, in (15.10.9) and (15.10.10) the symbols

a

and

b

are interchangeable. ►(c) If the parameter

c

in the differential equation equals

2-n=0,-1,-2,\dots

, then fundamental solutions in the neighborhood of

z=0

are given by

z^{n-1}

times those in (a) and (b), with

a

and

b

replaced throughout by

a+n-1

and

b+n-1

, respectively. … ►The

\genfrac{(}{)}{0.0pt}{}{6}{3}=20

connection formulas for the principal branches of Kummer’s solutions are: …

… ►

S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii), §15.12(iii).
Encodings:: BibTeX

… ►

FDLIBM (free C library)

ⓘ

Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer

G

-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

ⓘ

External Links:: Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §16.22.
Encodings:: BibTeX

►

J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer

G

-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt
Referenced by:: §16.22.
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…

… ►Tom M. Apostol (b. … ►Apostol was born on August 20, 1923. … ►He was a visiting professor at the University of Patras in Greece in 1978, and was elected a Corresponding Member of the Academy of Athens in 2001 (where he delivered his inaugural lecture in Greek). … ►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … ►He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

… ►

F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

ⓘ

Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

…

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Pearson (1968) tabulates $I_{x}\left(a,b\right)$ for $x=0.01(.01)1$ , $a,b=0.5(.5)11(1)50$ , with $b\leq a$ , to 7D.

►

•

Zhang and Jin (1996, Table 3.9) tabulates $I_{x}\left(a,b\right)$ for $x=0(.05)1$ , $a=0.5,1,3,5,10$ , $b=1,10$ to 8D.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

ⓘ

External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

ⓘ

External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

… ►

B. T. M. Murphy and A. D. Wood (1997) Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.

ⓘ

External Links:: ISSN 1073-2772, MathReview (W. Balser), ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

…

… ►

24.2.1

\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!},

|t|<2\pi

.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii), §26.14(iii)
Permalink:: http://dlmf.nist.gov/24.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

►

B_{2n+1}=0

,

… ►

24.2.3

\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!},

|t|<2\pi

.

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $n$ : integer, $t$ : real or complex and $x$ : real or complex
A&S Ref:: 23.1.1
Referenced by:: §18.2(xii), §24.4(iv), §24.4(vii)
Permalink:: http://dlmf.nist.gov/24.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

\widetilde{B}_{n}\left(x+1\right)=\widetilde{B}_{n}\left(x\right),

…

large%20a%20or%20b

21—30 of 67 matching pages

21: 1.12 Continued Fractions

22: 27.15 Chinese Remainder Theorem

23: Gergő Nemes

24: 15.10 Hypergeometric Differential Equation

25: Bibliography F

26: Tom M. Apostol

27: Bibliography S

28: 8.26 Tables

29: Bibliography M

30: 24.2 Definitions and Generating Functions