# Euler–Maclaurin formula

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##### 2: 25.2 Definition and Expansions
###### §25.2(iii) Representations by the Euler–MaclaurinFormula
25.2.8 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\,\mathrm{d}x,$ $\Re s>0$, $N=1,2,3,\dots$.
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
25.2.10 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$ $\Re s>-2n$, $n=1,2,3,\dots$.
##### 3: 2.10 Sums and Sequences
###### §2.10(i) Euler–MaclaurinFormula
This is the EulerMaclaurin formula. … In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\geq\frac{1}{2}(\alpha+1)$. For extensions of the EulerMaclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). …
##### 4: Bibliography E
• D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
• ##### 5: 25.11 Hurwitz Zeta Function
###### §25.11(iii) Representations by the Euler–MaclaurinFormula
25.11.5 $\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1% }-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\,\mathrm% {d}x,$ $s\neq 1$, $\Re s>0$, $a>0$, $N=0,1,2,3,\dots$.
25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$.
25.11.7 $\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+% \frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}% \int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}\,% \mathrm{d}x,$ $s\neq 1$, $a>0$, $n=1,2,3,\dots$, $\Re s>-2n$.
##### 6: Bibliography H
• M. Hauss (1997) An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta(2m+1)$ . Commun. Appl. Anal. 1 (1), pp. 15–32.
• ##### 7: Bibliography B
• B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
If $k$ in (3.5.4) is not arbitrarily large, and if odd-order derivatives of $f$ are known at the end points $a$ and $b$, then the composite trapezoidal rule can be improved by means of the EulerMaclaurin formula2.10(i)). …
##### 9: 18.17 Integrals
Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28). … Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in (Erdélyi et al., 1953b, §10.9(15)). … Some of the resulting formulas are given below. … Formulas (18.17.45) and (18.17.49) are integrated forms of the linearization formulas (18.18.22) and (18.18.23), respectively. …
##### 10: Errata
• Subsection 17.9(iii)

The title of the paragraph which was previously “Gasper’s $q$-Analog of Clausen’s Formula” has been changed to “Gasper’s $q$-Analog of Clausen’s Formula (16.12.2)”.

• Expansion

§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_{k}\left(x\right)$, $k\in\mathbb{Z}$, for the branches. The original two solutions are identified via $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

• Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

• Usability

Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

• Equation (5.11.8)

It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\operatorname{Ln}\Gamma\left(z+h\right)$ is valid for $h$ $(\in\mathbb{C})$; originally it was unnecessarily restricted to $[0,1]$.