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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:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Zeros of the Macdonald function of complex order. J. Comput. Appl. Math. 211 (2), pp. 223–231.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§10.42, §10.42.

Encodings:

BibTeX

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Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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22: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

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Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.

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External Links:

ISSN 0008-414X, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

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Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

Encodings:

BibTeX

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Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

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External Links:

Document, MathReview (P. Barrucand), ZentralBlatt

Referenced by:

§7.22(i), §7.25(iii).

Encodings:

BibTeX

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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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23: 7.23 Tables

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§7.23(ii) Real Variables

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§7.23(iii) Complex Variables, $z=x+\mathrm{i}y$

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Abramowitz and Stegun (1964, Chapter 7) includes $w\left(z\right)$, $x=0(.1)3.9$, $y=0(.1)3$, 6D.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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Fettis et al. (1973) gives the first 100 zeros of $\mathrm{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$, not for $\mathrm{erfc}z$), 11S.

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24: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

►If $a\ge -\frac{1}{2}$, then $U(a,x)$ has no real zeros. If , then $U(a,x)$ has no positive real zeros. If , $n=1,2,\mathrm{\dots }$, then $U(a,x)$ has $n$ positive real zeros. … ►For further information, including associated functions, see Olver (1959).

25: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

… ►where $a,b\in ℂ$ and . … ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►

§20.11(iv) Theta Functions with Characteristics

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26: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

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23.9.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{n=2}^{\mathrm{\infty }}{c}_n{z}^{2n-2},$$ ,

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.1

Referenced by:

§23.9, §23.9

Permalink:

http://dlmf.nist.gov/23.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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23.9.3 $$\zeta \left(z\right)=\frac{1}{z}-\sum _{n=2}^{\mathrm{\infty }}\frac{c_n}{2n-1}{z}^{2n-1},$$ .

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.5

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

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$c_2={\displaystyle \frac{1}{20}}{g}_2,$

… ►For $z\in ℂ$ …

27: Bibliography C

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Bessel functions $I_{\nu }(x)$ and $K_{\nu }(x)$ of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.

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Notes:

Double-precision Fortran, maximum accuracy 16S. For erratum see same journal 25 (1982), p. 207.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

2nd item.

Encodings:

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Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10 (1-2), pp. 13–26.

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External Links:

ISSN 1017-1398, MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii), §19.36(i), §19.36(i), §19.37(iv).

Encodings:

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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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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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Notes:

Complex interval standard function library. For details see Neher (2007)

External Links:

CoStLy

Referenced by:

§ ‣ Software Cross Index, M. Neher (2007).

Encodings:

BibTeX

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28: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

§30.9(ii) Oblate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►

§30.9(iii) Other Approximations and Expansions

… ►The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982). …

29: 19.36 Methods of Computation

… ►Because $U_{12}$ may be real and negative, or even complex, care is needed to ensure $\sqrt{x}=U_{12}$, and similarly for $y$ and $z$. … ►The function $\mathrm{el2}(x,k_c,a,b)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). Remedies for cancellation when $x$ is real and near 0 are supplied in Midy (1975). … ►

§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). …

30: 6.20 Approximations

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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 $\mathrm{Ei}\left(x\right)$, 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, pp. 41–42) gives Chebyshev expansions of $\mathrm{Ein}\left(ax\right)$, $\mathrm{Si}\left(ax\right)$, and $\mathrm{Cin}\left(ax\right)$ for $-1\le x\le 1$, $a\in ℂ$. The coefficients are given in terms of series of Bessel functions.

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