mathematical constants

SearchAdvancedHelp

(0.002 seconds)

21—30 of 37 matching pages

21: 10.24 Functions of Imaginary Order

… ►

10.24.3 $$\mathrm{\Gamma }\left(1+\mathrm{i}\nu \right)={\left(\frac{\pi \nu }{\sinh \left(\pi \nu \right)}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}{\gamma }_{\nu }},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Permalink:

http://dlmf.nist.gov/10.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

►where ${\gamma }_{\nu }$ is real and continuous with ${\gamma }_0=0$; compare (5.4.3). … ►

10.24.9 $${\tilde{Y}}_0\left(x\right)=Y_0\left(x\right)=\frac{2}{\pi }\left(\ln \left(\frac{1}{2}x\right)+\gamma \right)+O\left(x^2\ln x\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

►where $\gamma$ denotes Euler’s constant §5.2(ii). … ►For mathematical properties and applications of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …

22: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A more precise mathematical discussion then follows in §1.18(ix). … ►

§1.18(ix) Mathematical Background

… ► … ► ►

Self-adjoint extensions of a symmetric Operator

…

23: 5.19 Mathematical Applications

§5.19 Mathematical Applications

►

§5.19(i) Summation of Rational Functions

… ►

5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

… ►

$V={\displaystyle \frac{{\pi }^{\frac{1}{2}n}{r}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}},$

►

$S={\displaystyle \frac{2{\pi }^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{1}{2}n\right)}}={\displaystyle \frac{n}{r}}V.$

24: Bibliography P

… ►

Zonal Polynomials of Order $1$ Through $12$ . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

ⓘ

External Links:

MathReview

Referenced by:

§35.11.

Encodings:

BibTeX

… ►

Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§29.18(iv).

Encodings:

BibTeX

… ►

Complex Interval Arithmetic and its Applications. Mathematical Research, Vol. 105, Wiley-VCH Verlag Berlin GmbH, Berlin.

ⓘ

External Links:

ISBN 3-527-40134-2, MathReview (G. Alefeld), ZentralBlatt

Referenced by:

§3.1(ii).

Encodings:

BibTeX

… ►

Gauge Field Theories. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-26537-1, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§22.19(ii).

Encodings:

BibTeX

…

25: 31.17 Physical Applications

… ►

$\gamma =-2s,$

… ►More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …

26: 7.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$\delta$	arbitrary small positive constant.
$\gamma$	Euler’s constant (§5.2(ii)).

… ►The notations $P(z)$, $Q(z)$, and $\mathrm{\Phi }(z)$ are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions. …

27: Bibliography B

… ►

A guide to tables of Bessel functions. Mathematical Tables and Other Aids to Computation (now Mathematics of Computation) 1 (7), pp. 205–308.

ⓘ

External Links:

FullText, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§10.75(i).

Encodings:

BibTeX

… ►

Applied Combinatorial Mathematics. Robert E. Krieger, Malabar, FL.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§26.19, §26.20.

Encodings:

BibTeX

… ►

Emerging tools for experimental mathematics. Amer. Math. Monthly 106 (10), pp. 889–909.

ⓘ

External Links:

ISSN 0002-9890, MathReview (Michael M. Dediu), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

… ►

A Guide to Mathematical Tables: Supplement No. 1. Pergamon Press, New York.

ⓘ

Notes:

Supplement to “A Guide to Mathematical Tables” by A. V. Lebedev and R. M. Fedorova.

External Links:

MathReview

Referenced by:

A. V. Lebedev and R. M. Fedorova (1960).

Encodings:

BibTeX

… ►

The SLATEC Common Mathematical Library. In Sources and Development of Mathematical Software, W. R. Cowell (Ed.), pp. 302–320.

ⓘ

External Links:

GAMS (package SLATEC)

Referenced by:

SLATEC (free Fortran library).

Encodings:

BibTeX

…

28: Guide to Searching the DLMF

… ►In DLMF, you can search not only for text but also for mathematical expressions. … ►

Table 1: Query Examples

► ►►►►

Query	Matching records contain
…
Euler	the word ”Euler” or any of the various Euler terms such as Euler Gamma function $\mathrm{\Gamma }$, Euler Beta function $\mathrm{B}$, etc.
…
Gamma near/5 =	the terms $\mathrm{\Gamma }$ and = such that $\mathrm{\Gamma }$ is up to 5 terms before or after =.
…

►

…

29: Bibliography T

… ►

Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

ⓘ

External Links:

ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt

Referenced by:

§35.4(i), §35.9.

Encodings:

BibTeX

… ►

Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.

ⓘ

External Links:

ISBN 0-521-45718-1, MathReview (Stefan Kühnlein), ZentralBlatt

Referenced by:

§5.16.

Encodings:

BibTeX

… ►

Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

ⓘ

Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

Encodings:

BibTeX

… ►

Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

ⓘ

External Links:

ISBN 0-89871-361-7, MathReview (Moody T. Chu), ZentralBlatt

Referenced by:

§3.2(iii), §3.2(vii).

Encodings:

BibTeX

… ►

An Essay Toward a Unified Theory of Special Functions. Annals of Mathematics Studies, no. 18, Princeton University Press, Princeton, N.J..

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.5(ii).

Encodings:

BibTeX

…

30: 10.72 Mathematical Applications

§10.72 Mathematical Applications

… ►The number $m$ can also be replaced by any real constant $\lambda$ $(>-2)$ in the sense that ${(z-z_0)}^{-\lambda }$ $f(z)$ is analytic and nonvanishing at $z_0$; moreover, $g(z)$ is permitted to have a single or double pole at $z_0$. …