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11: 4.5 Inequalities

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12: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

… ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

13: 4.18 Inequalities

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14: 23.9 Laurent and Other Power Series

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15: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

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16: Bibliography E

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

Encodings:

BibTeX

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17: 25.11 Hurwitz Zeta Function

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§25.11(iii) Representations by the Euler–Maclaurin Formula

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25.11.5 $$\zeta (s,a)=\sum _{n=0}^N\frac{1}{{(n+a)}^s}+\frac{{(N+a)}^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{{(x+a)}^{s+1}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>0$, $a>0$, $N=0,1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Referenced by:

(25.11.7)

Permalink:

http://dlmf.nist.gov/25.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

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25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1985a, (25), p. 231)

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-05-08 by Clemens Heuberger

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

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25.11.7 $$\zeta (s,a)=\frac{1}{a^s}+\frac{1}{{(1+a)}^s}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}\frac{1}{{(1+a)}^{s+2k-1}}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{{(x+a)}^{s+2n+1}}dx,$$ $s\ne 1$, $a>0$, $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }s>-2n$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34).

Permalink:

http://dlmf.nist.gov/25.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

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18: 9.12 Scorer Functions

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§9.12(vi) Maclaurin Series

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19: 10.74 Methods of Computation

… ►Temme (1997) shows how to overcome this difficulty by use of the Maclaurin expansions for these coefficients or by use of auxiliary functions. …

20: 11.10 Anger–Weber Functions

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§11.10(iii) Maclaurin Series

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