艾里森山大学商业管理文凭证书【somewhat微aptao168】1949a

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1: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. ►

A Classification of Software

… ►

Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

►

Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

… ►

netlib

A collection of mathematical software, papers, and databases produced by the numerical analysis research community.

…

2: Jim Pitman

… ► 1949 in Tasmania) is a professor in the departments of statistics and mathematics at the University of California, Berkeley. …Pitman held a position in the Department of Mathematics and Mathematical Statistics at the University of Cambridge, England. …Pitman holds a B. …in statistics from the Australian National University, Canberra, and a Ph. … ►In November 2015, Pitman was named a Senior Associate Editor of the DLMF.

3: 7.8 Inequalities

… ►

7.8.2

\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+% (4/\pi)}},

x\geq 0

ⓘ

Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable
A&S Ref:: 7.1.13
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

►

7.8.3

\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)<\frac{1}{x+1},

x\geq 0

ⓘ

Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

… ►

7.8.6

\int_{0}^{x}e^{at^{2}}\,\mathrm{d}t<\frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2% \right),

a,x>0

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

►

7.8.7

\frac{\sinh x^{2}}{x}<{\mathrm{e}}^{x^{2}}F\left(x\right)=\int_{0}^{x}{\mathrm% {e}}^{t^{2}}\,\mathrm{d}t<\frac{{\mathrm{e}}^{x^{2}}-1}{x},

x>0

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable
Referenced by:: Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/7.8.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: A new inequality was added.
See also:: Annotations for §7.8 and Ch.7

… ►

7.8.8

\operatorname{erf}x<\sqrt{1-{\mathrm{e}}^{-4x^{2}/\pi}},

x>0

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm and $x$ : real variable
Source:: Pólya (1949, (1.5))
Referenced by:: §7.8, Erratum (V1.0.17) for Equation (7.8.8)
Permalink:: http://dlmf.nist.gov/7.8.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.17):: This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).
Suggested by Roberto Iacono
See also:: Annotations for §7.8 and Ch.7

4: 27.21 Tables

… ►Glaisher (1940) contains four tables: Table I tabulates, for all

n\leq 10^{4}

: (a) the canonical factorization of

n

into powers of primes; (b) the Euler totient

\phi\left(n\right)

; (c) the divisor function

d\left(n\right)

; (d) the sum

\sigma(n)

of these divisors. …7 of Abramowitz and Stegun (1964) also lists the factorizations in Glaisher’s Table I(a); Table 24. … ►Tables of the Ramanujan function

\tau\left(n\right)

are published in Lehmer (1943) and Watson (1949). Lehmer (1941) gives a comprehensive account of tables in the theory of numbers, including virtually every table published from 1918 to 1941. …Lehmer (1941) also has a section that supplies errata and corrections to all tables cited. …

5: 4.10 Integrals

… ►The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). … ►For

a,b\neq 0

, ►

4.10.8

\int e^{az}\,\mathrm{d}z=\frac{e^{az}}{a},

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $z$ : complex variable and $a\neq 0$ : complex
A&S Ref:: 4.2.54
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.10(ii), §4.10 and Ch.4

… ►

4.10.10

\int\frac{e^{az}-1}{e^{az}+1}\,\mathrm{d}z=\frac{2}{a}\ln\left(e^{az/2}+e^{-az% /2}\right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $a\neq 0$ : complex
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.10(ii), §4.10 and Ch.4

… ►Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).

6: Bibliography H

… ►

P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

ⓘ

External Links:: MathReview (C. Tanaka)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

W. Hahn (1949) Über Orthogonalpolynome, die

q

-Differenzengleichungen genügen. Math. Nachr. 2, pp. 4–34 (German).

ⓘ

External Links:: MathReview (N. J. Fine), ZentralBlatt
Referenced by:: §18.27(i).
Encodings:: BibTeX

… ►

G. H. Hardy (1949) Divergent Series. Clarendon Press, Oxford.

ⓘ

Notes:: Reprinted by the American Mathematical Society in 2000.
External Links:: MathReview (R. P. Agnew), ZentralBlatt
Referenced by:: §1.15(ii), §1.15(viii).
Encodings:: BibTeX

… ►

M. H. Hirata (1975) Flow near the bow of a steadily turning ship. J. Fluid Mech. 71 (2), pp. 283–291.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

N. J. Hitchin (2003) A lecture on the octahedron. Bull. London Math. Soc. 35 (5), pp. 577–600.

ⓘ

External Links:: ISSN 0024-6093, Document, MathReview (I. Dolgachev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

…

7: Bibliography P

… ►

P. Painlevé (1906) Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §32.2(iv).
Encodings:: BibTeX

… ►

A. R. Paterson (1983) A First Course in Fluid Dynamics. Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-25416-7; 0-521-27424-9, MathReview, ZentralBlatt
Referenced by:: §18.39(v).
Encodings:: BibTeX

… ►

T. Pearcey (1946) The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

ⓘ

External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: §36.2(ii).
Encodings:: BibTeX

… ►

M. Petkovšek, H. S. Wilf, and D. Zeilberger (1996)

A=B

. A K Peters Ltd., Wellesley, MA.

ⓘ

Notes:: With a separately available computer disk
External Links:: ISBN 1-56881-063-6, MathReview (Peter Paule), ZentralBlatt
Referenced by:: §16.23(iv).
Encodings:: BibTeX

… ►

G. Pólya (1949) Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

ⓘ

External Links:: MathReview (L. A. Aroian), ZentralBlatt
Referenced by:: (7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).
Encodings:: BibTeX

…

8: Bibliography T

… ►

A. Takemura (1984) 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

►

J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

ⓘ

Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

F. G. Tricomi (1949) Sul comportamento asintotico dell’

n

-esimo polinomio di Laguerre nell’intorno dell’ascissa

4n

. Comment. Math. Helv. 22, pp. 150–167.

ⓘ

External Links:: ISSN 0010-2571, Document, MathReview (G. Szegő), ZentralBlatt
Referenced by:: §18.16(iv).
Encodings:: BibTeX

… ►

A. Tucker (2006) Applied Combinatorics. 5th edition, John Wiley and Sons, New York.

ⓘ

Referenced by:: §26.15.
Encodings:: BibTeX

►

S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §9.1, Notations E, Notations E.
Encodings:: BibTeX

…

9: 4.40 Integrals

… ►

4.40.7

\int_{0}^{\infty}e^{-x}\frac{\sin\left(ax\right)}{\sinh x}\,\mathrm{d}x=\tfrac% {1}{2}\pi\coth\left(\tfrac{1}{2}\pi a\right)-\frac{1}{a},

a\neq 0

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.8

\int_{0}^{\infty}\frac{\sinh\left(ax\right)}{\sinh\left(\pi x\right)}\,\mathrm% {d}x=\tfrac{1}{2}\tan\left(\tfrac{1}{2}a\right),

-\pi<a<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\tan\NVar{z}$ : tangent function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.9

\int_{-\infty}^{\infty}\frac{e^{ax}}{\left(\cosh\left(\tfrac{1}{2}x\right)% \right)^{2}}\,\mathrm{d}x=\frac{4\pi a}{\sin\left(\pi a\right)},

-1<a<1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

►

4.40.10

{\int_{0}^{\infty}\frac{\tanh\left(ax\right)-\tanh\left(bx\right)}{x}\,\mathrm% {d}x=\ln\left(\frac{a}{b}\right)},

a>0

b>0

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.40.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(iii), §4.40 and Ch.4

… ►Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).

10: Bibliography W

… ►

J. Walker (1989) A drop of water becomes a gateway into the world of catastrophe optics. Scientific American 261, pp. 120–123.

ⓘ

Referenced by:: §36.14(ii).
Encodings:: BibTeX

… ►

G. N. Watson (1949) A table of Ramanujan’s function

\tau(n)

. Proc. London Math. Soc. (2) 51, pp. 1–13.

ⓘ

External Links:: Document, MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §27.21.
Encodings:: BibTeX

… ►

R. L. Wiegel (1960) A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7 (2), pp. 273–286.

ⓘ

External Links:: Document, MathReview (T. B. Benjamin), ZentralBlatt
Referenced by:: §21.9.
Encodings:: BibTeX

… ►

J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

ⓘ

External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §13.23(iv).
Encodings:: BibTeX

►

J. Wimp (1965) On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.

ⓘ

External Links:: FullText, Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §13.9(ii).
Encodings:: BibTeX

…