DLMF: Untitled Document

… ►

1.7.8

\min(a_{1},a_{2},\dots,a_{n})\leq M(r)\leq\max(a_{1},a_{2},\dots,a_{n}),

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Symbols:: $n$ : nonnegative integer and $M(\NVar{r})$ : weighted mean
A&S Ref:: 3.2.2
Permalink:: http://dlmf.nist.gov/1.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(iii), §1.7 and Ch.1

…

… ►Let

\mu=\max{(\nu-m,0)}

. …

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36.7.7

n_{\max}(m)=\left\lfloor\tfrac{256}{13}m-\tfrac{269}{52}\right\rfloor.

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Symbols:: $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $n$ : integer and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(iii), §36.7 and Ch.36

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S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.

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Notes:: Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.
External Links:: ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: 6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ 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, § ‣ Software Cross Index.
Encodings:: BibTeX

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… ►Equation (16.8.3) is of order

\max(p,q+1)

. … ►More generally if

z_{0}

(

\in\mathbb{C}

) is an arbitrary constant,

|z-z_{0}|>\max{(|z_{0}|,|z_{0}-1|)}

, and

|\operatorname{ph}\left(z_{0}-z\right)|<\pi

, then …

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A. J. MacLeod (1996a) Algorithm 757: MISCFUN, a software package to compute uncommon special functions. ACM Trans. Math. Software 22 (3), pp. 288–301.

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Notes:: Double-precision Fortran, maximum accuracy 20D.
External Links:: ISSN 0098-3500, Document, GAMS (module 757; package TOMS), MathReview Entry, ZentralBlatt
Referenced by:: 5th item, 1st item, 1st item, 4th item, 1st item, 1st item.
Encodings:: BibTeX

►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

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J. N. Merner (1962) Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

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Notes:: Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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N. Michel (2007) Precise Coulomb wave functions for a wide range of complex

\ell

,

\eta

and

z

. Computer Physics Communications 176 (3), pp. 232–249.

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Notes:: C++ code. Maximum accuracy 10D
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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J. Morris (1969) Algorithm 346: F-test probabilities [S14]. Comm. ACM 12 (3), pp. 184–185.

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Notes:: Includes Algol program, maximum accuracy 12S. For a remark see ACM Trans. Math. Software 14(1988), pp. 288-289
External Links:: ISSN 0001-0782, Document, Remark
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

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K. S. Kölbig (1981) A Program for Computing the Conical Functions of the First Kind

P^{m}_{-1/2+i\tau}(x)

for

m=0

and

m=1

. Comput. Phys. Comm. 23 (1), pp. 51–61.

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Notes:: Single-precision Fortran, maximum accuracy 15D.
External Links:: Document, Code
Referenced by:: §14.31(ii), 1st item.
Encodings:: BibTeX

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P. Kravanja, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos (1998) ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Notes:: Double-precision Fortran, maximum accuracy 25S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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D. A. Levine (1969) Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

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Notes:: Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.
External Links:: ISSN 0001-0782, Document, Remark 1 and Remark 2
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

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… ►

13.4.5

U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a\right)\Gamma\left(1+a-b\right% )}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^{-t}t^{a-1}}{t+z}\,\mathrm{d}t,

|\operatorname{ph}{z}|<\pi

,

\Re a>\max\left(\Re b-1,0\right)

,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part and $z$ : complex variable
Referenced by:: §13.10(vi), §13.4(i), (9.10.18)
Permalink:: http://dlmf.nist.gov/13.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(i), §13.4 and Ch.13

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13.4.7

U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma\left(a\right)% \Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2% }}K_{b-1}\left(2\sqrt{zt}\right)\,\mathrm{d}t,

\Re a>\max\left(\Re b-1,0\right)

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §13.4(i), §13.4 and Ch.13

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… ►

B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:: Single- and double-precision Fortran, maximum accuracy 32S.
External Links:: ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, 2nd item, 1st item.
Encodings:: BibTeX

… ►

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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41—50 of 75 matching pages

41: 1.7 Inequalities

42: 29.5 Special Cases and Limiting Forms

43: 36.7 Zeros

44: Bibliography Z

45: 16.8 Differential Equations

46: Bibliography M

47: Bibliography K

48: Bibliography L

49: 13.4 Integral Representations

50: Bibliography F