DLMF: Untitled Document

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19.36.2

1-\tfrac{3}{14}E_{2}+\tfrac{1}{6}E_{3}+\tfrac{9}{88}E_{2}^{2}-\tfrac{3}{22}E_{% 4}-\tfrac{9}{52}E_{2}E_{3}+\tfrac{3}{26}E_{5}-\tfrac{1}{16}E_{2}^{3}+\tfrac{3}% {40}E_{3}^{2}+\tfrac{3}{20}E_{2}E_{4}+\tfrac{45}{272}E_{2}^{2}E_{3}-\tfrac{9}{% 68}(E_{3}E_{4}+E_{2}E_{5}).

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Symbols:: $E_{s}(\mathbf{z})$ : symmetric function
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.36.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.36(i), §19.36 and Ch.19

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FDLIBM (free C library)

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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:: BibTeX

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§25.5 Integral Representations

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§25.5(i) In Terms of Elementary Functions

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§25.5(ii) In Terms of Other Functions

… ►For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). … ►

§25.5(iii) Contour Integrals

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§7.24(i) Approximations in Terms of Elementary Functions

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Hastings (1955) gives several minimax polynomial and rational approximations for $\operatorname{erf}x$ , $\operatorname{erfc}x$ and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ .

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\geq 3$ (15D).

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§6.20(i) Approximations in Terms of Elementary Functions

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$ -function (§13.2(i)) from which Chebyshev expansions near infinity for $E_{1}\left(z\right)$ , $\mathrm{f}\left(z\right)$ , and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

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T. M. MacRobert (1967) 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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S. M. Markov (1981) 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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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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

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X. Merrheim (1994) 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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J. Muller (1997) 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

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

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W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:: MathReview (H. O. Pollak), ZentralBlatt
Referenced by:: §5.6(i), §6.8, §7.8, §8.10.
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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

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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

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A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:: Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.
External Links:: ZentralBlatt
Referenced by:: §19.12, Ch.19, A. Cayley (1961).
Encodings:: BibTeX

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A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:: Second edition of Cayley (1895), unabridged and corrected
External Links:: ZentralBlatt
Referenced by:: §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).
Encodings:: BibTeX

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W. J. Cody and W. Waite (1980) Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.

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External Links:: ISBN 0138220646, ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

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W. J. Cody (1993a) Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.

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External Links:: ISSN 0098-3500, Document, GAMS (module 714; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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J. Igusa (1972) Theta Functions. Springer-Verlag, New York.

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Notes:: Die Grundlehren der mathematischen Wissenschaften, Band 194
External Links:: ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt
Referenced by:: §21.5(ii), §21.8, Ch.21.
Encodings:: BibTeX

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IMSL (commercial C, Fortran, and Java libraries) 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:: BibTeX

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E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

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External Links:: ZentralBlatt
Referenced by:: 4th item, 2nd item.
Encodings:: BibTeX

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E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.15(ii), 1st item, §29.7(i).
Encodings:: BibTeX

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K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

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… ►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)=\operatorname{Li}_{s}\left(1\right)

. … ►Further properties include …and … ►In terms of polylogarithms …

elementary%20functions

1—10 of 15 matching pages

1: 19.36 Methods of Computation

2: Bibliography F

3: 25.5 Integral Representations

§25.5 Integral Representations

§25.5(i) In Terms of Elementary Functions

§25.5(ii) In Terms of Other Functions

§25.5(iii) Contour Integrals

4: 7.24 Approximations

§7.24(i) Approximations in Terms of Elementary Functions

5: 6.20 Approximations

§6.20(i) Approximations in Terms of Elementary Functions

6: Bibliography M

7: Bibliography G

8: Bibliography C

9: Bibliography I

10: 25.12 Polylogarithms