relation%20to%20infinite%20partial%20fractions

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11: 6.16 Mathematical Applications

… ►The

n

th partial sum is given by …Hence, if

x

is fixed and

n\to\infty

, then

S_{n}(x)\to\frac{1}{4}\pi

0

, or

-\frac{1}{4}\pi

according as

0<x<\pi

x=0

, or

-\pi<x<0

; compare (6.2.14). … ►Hence if

x=\pi/(2n)

and

n\to\infty

, then the limiting value of

S_{n}(x)

overshoots

\frac{1}{4}\pi

by approximately 18%. … ►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then …where

\pi(x)

is the number of primes less than or equal to

x

. …

12: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was also a coauthor of three textbooks written to accompany the physics telecourse The Mechanical Universe …and Beyond. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … Ford Award, given to recognize authors of articles of expository excellence. … ►

25 Zeta and Related Functions

…

13: Bibliography I

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions,

q

-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

ⓘ

External Links:: ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

►

M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

… ►

A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

ⓘ

Notes:: In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207
External Links:: ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

…

14: Bibliography S

… ►

K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

… ►

A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

… ►

J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

ⓘ

Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

ⓘ

Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

…

15: 26.3 Lattice Paths: Binomial Coefficients

… ►

§26.3(i) Definitions

►

\genfrac{(}{)}{0.0pt}{}{m}{n}

is the number of ways of choosing

n

objects from a collection of

m

distinct objects without regard to order.

\genfrac{(}{)}{0.0pt}{}{m+n}{n}

is the number of lattice paths from

(0,0)

(m,n)

. …The number of lattice paths from

(0,0)

(m,n)

m\leq n

, that stay on or above the line

y=x

\genfrac{(}{)}{0.0pt}{}{m+n}{m}-\genfrac{(}{)}{0.0pt}{}{m+n}{m-1}.

… ►

§26.3(iii) Recurrence Relations

…

16: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

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

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

…

17: 26.5 Lattice Paths: Catalan Numbers

… ►

§26.5(i) Definitions

… ►It counts the number of lattice paths from

(0,0)

(n,n)

that stay on or above the line

y=x

. … ►

§26.5(iii) Recurrence Relations

… ►

26.5.6

C\left(n\right)\sim\frac{4^{n}}{\sqrt{\pi n^{3}}},

n\to\infty

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

►

26.5.7

\lim_{n\to\infty}\frac{C\left(n+1\right)}{C\left(n\right)}=4.

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.5(iv), §26.5 and Ch.26

18: 8.17 Incomplete Beta Functions

… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of

a

b

, and

x

, and also to complex values. … ►

§8.17(ii) Hypergeometric Representations

… ►

§8.17(iv) Recurrence Relations

… ►For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62). ►

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

…

19: 25.6 Integer Arguments

… ►

§25.6(i) Function Values

… ►

25.6.3

\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1},

n=1,2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $n$ : nonnegative integer
Source:: Apostol (1976, (20), p. 266); with $n\neq 0$
A&S Ref:: 23.2.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/25.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.8

\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{% 2k}{k}}.

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $k$ : nonnegative integer
Keywords:: infinite series
Source:: van der Poorten (1980, (2), p. 271)
Permalink:: http://dlmf.nist.gov/25.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.12

\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $\pi$ : the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Apostol (1985a, pp. 226, 227, 229)
Referenced by:: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §25.6(ii), §25.6 and Ch.25

… ►For related results see Basu and Apostol (2000).

20: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

§26.4(i) Definitions

… ►It is also the number of

k

-dimensional lattice paths from

(0,0,\ldots,0)

(n_{1},n_{2},\ldots,n_{k})

. For

k=0,1

, the multinomial coefficient is defined to be

1

. … ►(The empty set is considered to have one permutation consisting of no cycles.) … ►

§26.4(iii) Recurrence Relation

…