# Euler–Maclaurin formula

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##### 2: 25.2 Definition and Expansions
###### §25.2(iii) Representations by the Euler–MaclaurinFormula
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
##### 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)). …
where the integration contour separates the poles of $\Gamma\left(\tfrac{1}{3}+\tfrac{1}{3}t\right)$ from those of $\Gamma\left(-t\right)$. … For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … where $\gamma$ is Euler’s constant (§5.2(ii)). …
The first two standard solutions are: … Although $M\left(a,b,z\right)$ does not exist when $b=-n$, $n=0,1,2,\dots$, many formulas containing $M\left(a,b,z\right)$ continue to apply in their limiting form. …
13.2.18 $U\left(a,b,z\right)=\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}% +\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z^{2-\Re b}% \right),$ $1\leq\Re b<2$, $b\not=1$,