elementary functions

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31: 22.6 Elementary Identities

§22.6 Elementary Identities

… ►See §22.17.

32: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►With the upper sign in (12.10.2), expansions can be constructed for large $\mu$ in terms of elementary functions that are uniform for $t\in (-\mathrm{\infty },\mathrm{\infty })$ (§2.8(ii)). … ►The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). … ►

§12.10(ii) Negative $a$,

… ►

12.10.30 $${\overline{v}}_s(t)={\mathrm{i}}^s{v}_s(-\mathrm{i}t).$$

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Defines:

${\overline{v}}_s(t)$: solution (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $s$: nonnegative integer and $v_s(t)$: solution

Permalink:

http://dlmf.nist.gov/12.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(v), §12.10 and Ch.12

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§12.10(vi) Modifications of Expansions in Elementary Functions

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33: Guide to Searching the DLMF

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Table 3: A sample of recognized symbols

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Symbols	Comments
…
All elementary functions	Such as sin, cos, tan, Ln, log, exp
…

►

…

34: 19.15 Advantages of Symmetry

… ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). …

35: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.

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External Links:

ISSN 0366-8681, MathReview (David F. Griffiths), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

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The computation of elementary functions in radix $2^p$ . Computing 53 (3-4), pp. 219–232.

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Notes:

International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (Vienna, 1993)

External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

… ►

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Notes:

MPFR is a C library for multiple-precision floating-point computations with correct rounding. The main goal of MPFR is to provide a library for multiple-precision floating-point computation which is both efficient and has well-defined semantics. MPFR includes most of the elementary functions. It also implements the gamma function, and more special functions are planned in the future.

External Links:

MPFR

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.

Encodings:

BibTeX

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Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.

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Notes:

A 2nd edition was published in 2006.

External Links:

ISBN 0-8176-3990-X, MathReview (Warren E. Ferguson, Jr.), ZentralBlatt

Referenced by:

§4.45(i).

Encodings:

BibTeX

…

36: 28.8 Asymptotic Expansions for Large $q$

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

37: Bibliography L

… ►

Evaluating elementary functions with guaranteed precision. Programming and Computer Software 27 (2), pp. 101–110.

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External Links:

ISSN 0361-7688, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(ii).

Encodings:

BibTeX

… ►

Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.

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External Links:

ISSN 0006-3835, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i), §4.46.

Encodings:

BibTeX

…

38: 33.20 Expansions for Small $|ϵ|$

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

39: Bibliography P

… ►

Integrals and Series: Elementary Functions, Vol. 1. Gordon & Breach Science Publishers, New York.

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Notes:

Translated from the Russian and with a preface by N. M. Queen. Table errata: Math. Comp. v. 66 (1997), no. 220, pp. 1765–1766, Math. Comp. v. 65 (1996), no. 215, pp. 1380–1381 .

External Links:

ISBN 2-88124-097-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §1.8(v), §24.7(iii), §4.10(iii), §4.11, §4.26(v), §4.27, §4.40(v), §4.41.

Encodings:

BibTeX

…

40: 7.7 Integral Representations

… ►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. …