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21: Bibliography T

… ►

G. ’t Hooft and M. Veltman (1979) Scalar one-loop integrals. Nuclear Phys. B 153 (3-4), pp. 365–401.

ⓘ

External Links:: ISSN 0029-5582, Document, MathReview (L. O’Raifeartaigh)
Referenced by:: §25.12(i).
Encodings:: BibTeX

… ►

N. M. Temme (1994b) Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.
Encodings:: BibTeX

… ►

N.M. Temme and E.J.M. Veling (2022) Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z. Indagationes Mathematicae.

ⓘ

External Links:: ISSN 0019-3577, Document, Link
Referenced by:: §13.8(iv).
Encodings:: BibTeX

… ►

W. J. Thompson (1997) Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.

ⓘ

Notes:: With CD-ROM containing a large collection of mathematical function software written in Fortran 90 and Mathematica (an edition with the same software in C and Mathematica exists also). The functions are computed for real variables only. Maximum accuracy 12D.
External Links:: ISBN 0-471-18171-4, MathReview, ZentralBlatt
Referenced by:: §19.36(ii), § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

P. G. Todorov (1978) Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

…

22: 10.72 Mathematical Applications

… ►where

z

is a real or complex variable and

u

is a large real or complex parameter. … ►In regions in which (10.72.1) has a simple turning point

z_{0}

, that is,

f(z)

and

g(z)

are analytic (or with weaker conditions if

z=x

is a real variable) and

z_{0}

is a simple zero of

f(z)

, asymptotic expansions of the solutions

w

for large

u

can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order

\tfrac{1}{3}

(§9.6(i)). … ►In regions in which the function

f(z)

has a simple pole at

z=z_{0}

and

(z-z_{0})^{2}g(z)

is analytic at

z=z_{0}

(the case

\lambda=-1

in §10.72(i)), asymptotic expansions of the solutions

w

of (10.72.1) for large

u

can be constructed in terms of Bessel functions and modified Bessel functions of order

\pm\sqrt{1+4\rho}

, where

\rho

is the limiting value of

(z-z_{0})^{2}g(z)

z\to z_{0}

. …

23: 27.16 Cryptography

… ►For example, a code maker chooses two large primes

p

and

q

of about 400 decimal digits each. … ►To code a message by this method, we replace each letter by two digits, say

A=01

B=02

\dots

Z=26

, and divide the message into pieces of convenient length smaller than the public value

n=pq

. …

24: 2.4 Contour Integrals

… ►Let

\mathscr{P}

denote the path for the contour integral … ►The coefficients

b_{s}

are determined as in §2.3(iii), the branch of

\operatorname{ph}p_{0}

being chosen to satisfy … ►Higher coefficients

b_{2s}

in (2.4.15) can be found from (2.3.18) with

\lambda=1

\mu=2

, and

s

replaced by

2s

. For integral representations of the

b_{2s}

and their asymptotic behavior as

s\to\infty

see Boyd (1995). … ►in which

z

is a large real or complex parameter,

p(\alpha,t)

and

q(\alpha,t)

are analytic functions of

t

and continuous in

t

and a second parameter

\alpha

. …

25: 19.36 Methods of Computation

… ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function

\operatorname{cel}\left(k_{c},p,a,b\right)

is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►Bulirsch (1969a, b) extend Bartky’s transformation to

\operatorname{el3}\left(x,k_{c},p\right)

by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …

26: Bibliography C

… ►

L. Carlitz (1960) Note on Nörlund’s polynomial

B_{n}^{(z)}

. Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

ⓘ

External Links:: FullText, MathReview (M. P. Drazin), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

ⓘ

External Links:: MathReview (J. A. Donaldson), ZentralBlatt
Referenced by:: §19.31.
Encodings:: BibTeX

… ►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

ⓘ

External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

ⓘ

External Links:: Document
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

Cunningham Project (website)

ⓘ

Notes:: Seeks to factor numbers of the form $b^{n}\pm 1$ for $b=2,3,5,6,7,10,11,12$ .
External Links:: Home Page
Referenced by:: 3rd item.
Encodings:: BibTeX

…

27: 12.11 Zeros

… ►When

a>-\frac{1}{2}

the zeros are asymptotically given by

z_{a,s}

and

\overline{z_{a,s}}

, where

s

is a large positive integer and … ►For large negative values of

a

the real zeros of

U\left(a,x\right)

U'\left(a,x\right)

V\left(a,x\right)

, and

V'\left(a,x\right)

can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

28: Bibliography G

… ►

M. Geller and E. W. Ng (1971) A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

ⓘ

Notes:: See for the first part same journal, Sect. B 73, 1-20 (1969)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

R. G. Gordon (1970) Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

ⓘ

External Links:: Document, MathReview (D. B. Fairlie)
Referenced by:: (9.12.22), (9.12.23), §9.12(vii), §9.17(iii).
Encodings:: BibTeX

… ►

H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

ⓘ

External Links:: FullText, MathReview (B. M. Stewart), ZentralBlatt
Referenced by:: §24.15(iii), §24.6.
Encodings:: BibTeX

… ►

GSL (free C library) GNU Scientific Library The GNU Project.

ⓘ

Notes:: The GNU Scientific Library is a large general purpose numerical software library with broad coverage of elementary and special functions. Implementation is in double precision.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

B. Guo (1998) Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

ⓘ

External Links:: ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt
Referenced by:: §18.38(i).
Encodings:: BibTeX

…

29: 17.2 Calculus

… ►Note that (17.2.6_1) is just (27.14.14) with

a=d=0

and

-b=c=1

. … ►

17.2.8

\frac{\left(a;q^{-1}\right)_{n}}{\left(b;q^{-1}\right)_{n}}=\frac{\left(a^{-1}% ;q\right)_{n}}{\left(b^{-1};q\right)_{n}}\left(\frac{a}{b}\right)^{n},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.10

\frac{\left(a;q\right)_{n}}{\left(b;q\right)_{n}}=\frac{\left(q^{1-n}/a;q% \right)_{n}}{\left(q^{1-n}/b;q\right)_{n}}\left(\frac{a}{b}\right)^{n},

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(i), §17.2 and Ch.17

… ►

17.2.12

\frac{\left(aq^{-n};q\right)_{n}}{\left(bq^{-n};q\right)_{n}}=\frac{\left(q/a;% q\right)_{n}}{\left(q/b;q\right)_{n}}\left(\frac{a}{b}\right)^{n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), $q$ : complex base and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/17.2.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §17.2(i), §17.2 and Ch.17

… ►These identities are the first in a large collection of similar results. …

30: 2.11 Remainder Terms; Stokes Phenomenon

… ►with

m

a large integer. … ►For large

\rho

the integrand has a saddle point at

t=e^{-i\theta}

. … ►For higher-order differential equations, see Olde Daalhuis (1998a, b). … ►For nonlinear differential equations see Olde Daalhuis (2005a, b). ►For another approach see Paris (2001a, b). …