大学结业证书是什么学历〖办证V信ATV1819〗16d

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1: 16 Generalized Hypergeometric Functions & Meijer G-Function

Chapter 16 Generalized Hypergeometric Functions and Meijer $G$-Function

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2: Staff

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Richard A. Askey, University of Wisconsin, Chaps. 1, 5, 16

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Adri B. Olde Daalhuis, University of Edinburgh, Chaps. 13, 15, 16

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Brian D. Sleeman, University of Leeds, Chap. 31

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Adri B. Olde Daalhuis, The University of Edinburgh, for Chaps. 13, 15, 16

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Brian D. Sleeman, University of Leeds, for Chap. 31 (deceased)

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3: 11.15 Approximations

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

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4: Publications

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Toward a Revised NBS Handbook of Mathematical Functions, Technical Report NISTIR 6072 (September 1997), National Institute of Standards and Technology.

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Design of a Digital Mathematical Library for Science, Technology and Education, Proceedings of the IEEE Forum on Research and Technology Advances in Digital Libraries (IEEE ADL ’99, Baltimore, Maryland, May 19, 1999).

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The DLMF Project: A New Initiative in Classical Special Functions, in Special Functions—Proceedings of the International Workshop, Hong Kong, June 21-25, 1999 (C. Dunkl, M. Ismail, R. Wong, eds.), World Scientific, pp. 207–220.

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Handbook of Mathematical Functions, in A Century of Excellence in Measurements Standards and Technology (D. R. Lide, ed.), CRC Press, pp. 135–139.

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Advances in Math Search, Proceedings 6th International Congress on Industrial and Applied Mathematics, Zürich, Switzerland, July 17, 2007.

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5: Bille C. Carlson

… ► 1924 in Cambridge, Massachusetts, d. …Department of Energy) at Iowa State University, Ames, Iowa, until his death on August 16, 2013. … ►In his paper Lauricella’s hypergeometric function $F_D$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …

6: 27.2 Functions

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27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

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Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. ►

27.2.10 $${\sigma }_{\alpha }\left(n\right)=\sum _{d|n}{d}^{\alpha },$$

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Defines:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number

Symbols:

$d$: positive integer, $n$: positive integer and $\alpha$: parameter

A&S Ref:

24.3.3 I.A

Referenced by:

§27.14(ii)

Permalink:

http://dlmf.nist.gov/27.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Note that ${\sigma }_0\left(n\right)=d\left(n\right)$. … ►Table 27.2.2 tabulates the Euler totient function $\varphi \left(n\right)$, the divisor function $d\left(n\right)$ ($={\sigma }_0(n)$), and the sum of the divisors $\sigma (n)$ ($={\sigma }_1(n)$), for $n=1(1)52$. …

7: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. …

8: 4.10 Integrals

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4.10.1 $$\int \frac{dz}{z}=\ln z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.48

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.2 $$\int \ln zdz=z\ln z-z,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.49

Permalink:

http://dlmf.nist.gov/4.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.4 $$\int \frac{dz}{z\ln z}=\ln \left(\ln z\right),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.52

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

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4.10.5 $${\int }_0^1\frac{\ln t}{1-t}dt=-\frac{{\pi }^2}{6},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.55

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §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).

9: Richard A. Askey

… ►Louis, Missouri, d. … ►D. … ►

16 Generalized Hypergeometric Functions & Meijer G-Function

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10: Bibliography

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Chapter $16$ of Ramanujan’s second notebook: Theta-functions and $q$-series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.

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External Links:

ISSN 0065-9266, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§20.11(ii).

Encodings:

BibTeX

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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions $I_{\nu }(x)$ and $J_{\nu }(x)$, $x\ge 0$, $\nu \ge 0$ . ACM Trans. Math. Software 3 (1), pp. 93–95.

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Notes:

Single-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0098-3500, Document, Companion paper. Erratum. GAMS (module 511; package TOMS)

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§17.16, Ch.17.

Encodings:

BibTeX

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Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

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External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

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