relation to elementary functions

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21: Bibliography F

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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:

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

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Guide to tables of elliptic functions. Math. Tables and Other Aids to Computation 3 (24), pp. 229–281.

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External Links:

FullText, MathReview (S. C. van Veen)

Referenced by:

§19.37(i).

Encodings:

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

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22: Bibliography H

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

External Links:

ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

Referenced by:

§1.2(vi).

Encodings:

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An Introduction to the Theory of Numbers. 5th edition, The Clarendon Press Oxford University Press, New York-Oxford.

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External Links:

ISBN 0-19-853170-2; 0-19-853171-0, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§20.12(i), Ch.27.

Encodings:

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An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

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External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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Wavelets: mathematical preliminaries. In Wavelets: an elementary treatment of theory and applications, Ser. Approx. Decompos., Vol. 1, pp. 13–26.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Tom H. Koornwinder.

Encodings:

BibTeX

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23: Bibliography B

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Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Praveen Agarwal), ZentralBlatt

Referenced by:

§14.17(iii), §14.17(iii).

Encodings:

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:

Document, MathReview (C. V. Coffman), ZentralBlatt

Referenced by:

§10.21(xi).

Encodings:

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Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach. 23 (2), pp. 242–251.

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External Links:

Document, MathReview (Amnon Barak), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

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24: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ► … ►The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … … ►SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …

25: 3.5 Quadrature

… ►The elementary trapezoidal rule is given by … ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … …

26: 25.12 Polylogarithms

… ►The cosine series in (25.12.7) has the elementary sum … ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …and … ►In terms of polylogarithms …

27: Bibliography R

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

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Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

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External Links:

ISBN 0-201-87073-8, MathReview (Norman J. Richert)

Referenced by:

§27.17.

Encodings:

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Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

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28: Bibliography I

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IMSL Nuerical Libraries..

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Notes:

The IMSL Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.

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.

Encodings:

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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External Links:

ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

Encodings:

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

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From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.

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External Links:

ISBN 3-528-06355-6, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§32.2(vi), §32.7(vii).

Encodings:

BibTeX

29: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►Define the elementary symmetric function $E_s(𝐳)$ by … ►This form of $T_N$ can be applied to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) if we use …The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. …

30: 1.8 Fourier Series

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§1.8(i) Definitions and Elementary Properties

… ►Here $c_n$ is related to $a_n$ and $b_n$ in (1.8.1), (1.8.2) by $c_n=\frac{1}{2}(a_n-\mathrm{i}{b}_n)$, $c_{-n}=\frac{1}{2}(a_n+\mathrm{i}{b}_n)$ for $n>0$ and $c_0=\frac{1}{2}{a}_0$. … ►As $n\to \mathrm{\infty }$ … ►Then the series (1.8.1) converges to the sum … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …