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21: 6.16 Mathematical Applications

… ►uniformly for $x\in [-\pi ,\pi ]$. … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

22: 25.12 Polylogarithms

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25.12.2 $${\mathrm{Li}}_2\left(z\right)=-{\int }_0^z{t}^{-1}\ln \left(1-t\right)dt,$$ $z\in ℂ\setminus (1,\mathrm{\infty })$.

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Symbols:

$ℂ$: complex plane, $dx$: differential of $x$ , ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\int$: integral, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

definite integral, integral representation

Source:

Maximon (2003, (2.1), p. 2807)

A&S Ref:

27.7.1 (is a modified form with $z=1-x$)

Referenced by:

§25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). … ►

25.12.3 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{z}{z-1}\right)=-\frac{1}{2}{(\ln \left(1-z\right))}^2,$$ $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

connection formula

Source:

Maximon (2003, (3.4), p. 2808)

A&S Ref:

27.7.5 (is a modified form with $z=1-x$)

Permalink:

http://dlmf.nist.gov/25.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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23: 5.11 Asymptotic Expansions

… ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►

5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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24: 36.5 Stokes Sets

… ►where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. … ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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25: Bibliography I

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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-0-7381-9720-3, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-1-5044-4602-0, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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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

…

26: 18.39 Applications in the Physical Sciences

… ►in which case the probability density is time-independent, as ${|\mathrm{\Psi }(x,t)|}^2=\mathrm{\Psi }(x,t)\overline{\mathrm{\Psi }(x,t)}={|{\psi }_n(x)|}^2$. … ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$. … ►which maps $ϵ\in [0,\mathrm{\infty })$ onto $x\in [-1,1]$. …

27: Bibliography

… ►

Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

… ►

On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

External Links:

Code, Document

Referenced by:

1st item.

Encodings:

BibTeX

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Introduction to Interval Computations. Computer Science and Applied Mathematics, Academic Press Inc., New York.

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Notes:

Translated from the German by Jon Rokne

External Links:

ISBN 0-12-049820-0, MathReview (H. Ratschek), ZentralBlatt

Referenced by:

§3.1(ii).

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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28: Errata

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The constraint $(x,y)\ne (0,0)$ was added to the first sentence of this section.

… ►

Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for $a_k(\nu )$ coefficient, Pochhammer symbol representation in (11.9.4) for $a_k(\mu ,\nu )$ coefficient, and (12.14.4_5).

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The correct corner coordinates for the 9-point square, given on the last line of this table, are $(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$. Originally they were given incorrectly as $(\pm \sqrt{\frac{3}{5}}h,0)$, $(\pm \sqrt{\frac{3}{5}}h,0)$.

Diagram	$(x_j,y_j)$	$w_j$	$R$
⋮
	$(0,0)$	$\frac{16}{81}$	$O\left(h^6\right)$
	$(\pm \sqrt{\frac{3}{5}}h,0)$, $(0,\pm \sqrt{\frac{3}{5}}h)$	$\frac{10}{81}$
	$(\pm \sqrt{\frac{3}{5}}h,\pm \sqrt{\frac{3}{5}}h)$	$\frac{25}{324}$

Reported 2014-01-13 by Stanley Oleszczuk.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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29: Bibliography C

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

Encodings:

BibTeX

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

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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30: 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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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Tables of the line broadening function $H(a,v)$ . Monthly Notices Roy. Astronom. Soc. 129, pp. 221–235.

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External Links:

FullText

Referenced by:

§7.19(i), 3rd 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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