for%20inverse%20function

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11—18 of 18 matching pages

11: Bibliography O

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A. B. Olde Daalhuis (2004a) Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

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External Links:: ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt
Referenced by:: §2.9(i).
Encodings:: BibTeX

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J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:: ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §13.20(iii), §13.20(iv).
Encodings:: BibTeX

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K. Ono (2000) Distribution of the partition function modulo

m

. Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:: ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §27.14(v).
Encodings:: BibTeX

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M. Onoe (1956) Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

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External Links:: ISSN 0891-6837, MathReview Entry, ZentralBlatt
Referenced by:: §10.29(i), §10.6(i).
Encodings:: BibTeX

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12: 26.14 Permutations: Order Notation

… ►As an example,

35247816

is an element of

\mathfrak{S}_{8}.

The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ►

§26.14(ii) Generating Functions

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26.14.4

\sum_{n,k=0}^{\infty}\genfrac{<}{>}{0.0pt}{}{n}{k}x^{k}\,\frac{t^{n}}{n!}=% \frac{1-x}{\exp((x-1)t)-x},

|x|<1,|t|<1

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $\exp\NVar{z}$ : exponential function, $!$ : factorial (as in $n!$ ), $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §26.14(iii)
Permalink:: http://dlmf.nist.gov/26.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(ii), §26.14 and Ch.26

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13: 26.13 Permutations: Cycle Notation

… ►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … ►

26.13.6

{\left(j,k\right)}={\left(k-1,k\right)}{\left(k-2,k-1\right)}\cdots{\left(j+1,% j+2\right)}\*{\left(j,j+1\right)}{\left(j+1,j+2\right)}\cdots{\left(k-1,k% \right)}.

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Symbols:: ${\left(\NVar{S}\right)}$ : cycle, $j$ : nonnegative integer and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.13 and Ch.26

… ►Given a permutation

\sigma\in\mathfrak{S}_{n}

, the inversion number of

\sigma

, denoted

\mathop{\mathrm{inv}}(\sigma)

, is the least number of adjacent transpositions required to represent

\sigma

. …

14: Bibliography B

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A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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W. G. Bickley and J. Nayler (1935) A short table of the functions

\mathrm{Ki}_{n}(x)

, from

n=1

n=16

. Phil. Mag. Series 7 20, pp. 343–347.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.43(iii), 2nd item.
Encodings:: BibTeX

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J. M. Blair, C. A. Edwards, and J. H. Johnson (1976) Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.17(iii), §7.17(iii).
Encodings:: BibTeX

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S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §35.5(i).
Encodings:: BibTeX

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15: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions

E_{s}

are defined by (19.19.4). … ►The function

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

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

§19.36(iii) Via Theta Functions

… ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

16: Bibliography W

… ►

E. L. Wachspress (2000) Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

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External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §22.20(ii), §22.20(v).
Encodings:: BibTeX

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P. L. Walker (1991) Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §4.12.
Encodings:: BibTeX

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P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:: Document
Referenced by:: (20.7.34), §20.7(ix).
Encodings:: BibTeX

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

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17: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). … ►

§12.10(vi) Modifications of Expansions in Elementary Functions

… ►Inversely, with

w=2^{-\frac{1}{3}}\zeta

, … ►

Modified Expansions

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18: Errata

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Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Figure 4.3.1

This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

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Equations (4.45.8), (4.45.9)

These equations have been rewritten to improve the numerical computation of $\operatorname{arctan}x$ .

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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