mathematical constants

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1: 3.12 Mathematical Constants

§3.12 Mathematical Constants

… ►For access to online high-precision numerical values of mathematical constants see Sloane (2003). …

2: Bibliography F

… ►

S. R. Finch (2003) Mathematical Constants. Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-81805-2, MathReview (E. J. Barbeau), ZentralBlatt
Referenced by:: §3.12.
Encodings:: BibTeX

…

3: 28.32 Mathematical Applications

… ►

4: 12.17 Physical Applications

… ►The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation ►

12.17.1

\nabla^{2}w+k^{2}w=0,

ⓘ

Symbols:: $k$ : constant
Referenced by:: §12.17
Permalink:: http://dlmf.nist.gov/12.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.17 and Ch.12

►where

k

is a constant, and

\nabla^{2}

is the Laplacian … ►

12.17.4

\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{2}w}{{\partial\xi}^{2}}+% \frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+\frac{{\partial}^{2}w}{{% \partial\zeta}^{2}}+k^{2}w=0.

ⓘ

Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $k$ : constant, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/12.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.17 and Ch.12

… ►with arbitrary constants

\sigma,\lambda

. …

5: Bibliography R

… ►

M. Reed and B. Simon (1975) Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

ⓘ

External Links:: ISBN 978-0-12-585002-5, MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

►

M. Reed and B. Simon (1978) Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

ⓘ

External Links:: ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

►

M. Reed and B. Simon (1979) Methods of Modern Mathematical Physics, Vol. 3, Scattering Theory. Academic Press, New York.

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External Links:: ISBN 0-12-585003-4 (vol.3), MathReview (P. R. Chernoff)
Referenced by:: §1.18(x).
Encodings:: BibTeX

►

M. Reed and B. Simon (1980) Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.

ⓘ

Notes:: revised and expanded Edition from 1972,
External Links:: ISBN 10: 0-12-585050-6(vol. 1), MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

W. Rudin (1976) Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:: International Series in Pure and Applied Mathematics
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.4(iii).
Encodings:: BibTeX

…

6: Bibliography

… ►

M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

… ►

J. A. Adam (2002) The mathematical physics of rainbows and glories. Phys. Rep. 356 (4-5), pp. 229–365 (English).

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External Links:: Document, FullText, ZentralBlatt
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

… ►

N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

… ►

G. B. Arfken and H. J. Weber (2005) Mathematical Methods for Physicists. 6th edition, Elsevier, Oxford.

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External Links:: ISBN 0-12-059876-0;0-12-088584-0, MathReview, ZentralBlatt
Referenced by:: §1.17(ii), §1.17(iii).
Encodings:: BibTeX

… ►

V. I. Arnol^′d (1997) Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Notes:: Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition
External Links:: ISBN 0-387-96890-3, MathReview
Referenced by:: §21.5(i).
Encodings:: BibTeX

…

7: 31.16 Mathematical Applications

§31.16 Mathematical Applications

… ►

31.16.5

P_{j}=\frac{(\epsilon-j+n)j(\beta+j-1)(\gamma+\delta+j-2)}{(\gamma+\delta+2j-3% )(\gamma+\delta+2j-2)},

ⓘ

Defines:: $P_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.6

Q_{j}=-aj(j+\gamma+\delta-1)-q+\frac{(j-n)(j+\beta)(j+\gamma)(j+\gamma+\delta-% 1)}{(2j+\gamma+\delta)(2j+\gamma+\delta-1)}+\frac{(j+n+\gamma+\delta-1)j(j+% \delta-1)(j-\beta+\gamma+\delta-1)}{(2j+\gamma+\delta-1)(2j+\gamma+\delta-2)},

ⓘ

Defines:: $Q_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.16.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.7

R_{j}=\frac{(n-j)(j+n+\gamma+\delta)(j+\gamma)(j+\delta)}{(\gamma+\delta+2j)(% \gamma+\delta+2j+1)}.

ⓘ

Defines:: $R_{j}$ (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $j$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/31.16.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.16(ii), §31.16 and Ch.31

… ►

8: 8.22 Mathematical Applications

§8.22 Mathematical Applications

… ►

8.22.1

F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $F_{\NVar{p}}\left(\NVar{z}\right)$ : terminant function, $z$ : complex variable and $p$ : parameter
Permalink:: http://dlmf.nist.gov/8.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §8.22(i), §8.22 and Ch.8

… ►The function

\Gamma\left(a,z\right)

, with

|\operatorname{ph}a|\leq\tfrac{1}{2}\pi

and

\operatorname{ph}z=\tfrac{1}{2}\pi

, has an intimate connection with the Riemann zeta function

\zeta\left(s\right)

(§25.2(i)) on the critical line

\Re s=\tfrac{1}{2}

. … ►

8.22.2

\zeta_{x}(s)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{x}\frac{t^{s-1}}{e^{t}-1}% \,\mathrm{d}t,

\Re s>1

ⓘ

Defines:: $\zeta_{x}(s)$ : incomplete Riemann zeta function (locally)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.22.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §8.22(ii), §8.22 and Ch.8

…

9: Bibliography M

… ►

H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:: Document, MathReview (K.-B. Gundlach), ZentralBlatt
Referenced by:: §35.4(ii).
Encodings:: BibTeX

… ►

Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:: A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.
External Links:: Magma
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

J. Meixner, F. W. Schäfke, and G. Wolf (1980) Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies. Lecture Notes in Mathematics, Vol. 837, Springer-Verlag, Berlin-New York.

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External Links:: ISBN 3-540-10282-5, 0-387-10282-5, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §28.10(ii), §28.11, §28.2(v), §28.27, §28.28(i), §28.28(v), §28.30(ii), §28.6(i), §28.6(i), §28.6(i), §28.7, §28.7, Ch.28, §30.11(vi), §30.12, §30.13(v), §30.14(v), §30.15(i), §30.3(iv), Ch.30, Profile Gerhard Wolf.
Encodings:: BibTeX

… ►

R. E. Moore (1979) Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-161-4, MathReview (H. Ratschek), ZentralBlatt
Referenced by:: §3.1(ii).
Encodings:: BibTeX

… ►

S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.

ⓘ

Notes:: Includes diskette
External Links:: ISBN 0-13-578998-2, MathReview, ZentralBlatt
Referenced by:: 2nd item, CEPHES (free C library).
Encodings:: BibTeX

…

10: 13.27 Mathematical Applications

§13.27 Mathematical Applications

… ►

13.27.1

g=\begin{pmatrix}1&\alpha&\beta\\ 0&\gamma&\delta\\ 0&0&1\end{pmatrix},

ⓘ

Permalink:: http://dlmf.nist.gov/13.27.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.27 and Ch.13

►where

\alpha

\beta

\gamma

\delta

are real numbers, and

\gamma>0

. …