In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823 ) and Giovanni Antonio Amedeo Plana (1820 ). It states that
[ 1]
∑
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f
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{\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx}
For the case
a
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0
{\displaystyle a=0}
∑
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f
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{\displaystyle \sum _{n=0}^{\infty }f(n)={\frac {1}{2}}f(0)+\int _{0}^{\infty }f(x)\,dx+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.}
It holds for functions ƒ that are holomorphic in the region Re(z ) ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that |ƒ | is bounded by C /|z |1+ε in this region for some constants C , ε > 0, though the formula also holds under much weaker bounds. (Olver 1997 , p.290).
An example is provided by the Hurwitz zeta function ,
ζ
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1
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1
−
s
s
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1
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α
s
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∫
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∞
sin
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arctan
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{\displaystyle \zeta (s,\alpha )=\sum _{n=0}^{\infty }{\frac {1}{(n+\alpha )^{s}}}={\frac {\alpha ^{1-s}}{s-1}}+{\frac {1}{2\alpha ^{s}}}+2\int _{0}^{\infty }{\frac {\sin \left(s\arctan {\frac {t}{\alpha }}\right)}{(\alpha ^{2}+t^{2})^{\frac {s}{2}}}}{\frac {dt}{e^{2\pi t}-1}},}
which holds for all
s
∈
C
{\displaystyle s\in \mathbb {C} }
s ≠ 1
e
−
n
n
x
{\displaystyle e^{-n}n^{x}}
Γ
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x
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1
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=
Li
−
x
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e
−
1
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+
θ
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x
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{\displaystyle \Gamma (x+1)=\operatorname {Li} _{-x}\left(e^{-1}\right)+\theta (x)}
Γ
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x
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{\displaystyle \Gamma (x)}
gamma function ,
Li
s
(
z
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{\displaystyle \operatorname {Li} _{s}\left(z\right)}
polylogarithm and
θ
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∫
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∞
2
t
x
e
2
π
t
−
1
sin
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π
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2
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d
t
{\displaystyle \theta (x)=\int _{0}^{\infty }{\frac {2t^{x}}{e^{2\pi t}-1}}\sin \left({\frac {\pi x}{2}}-t\right)dt}
Abel also gave the following variation for alternating sums:
∑
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∞
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−
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f
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f
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∫
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∞
f
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−
f
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2
sinh
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π
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d
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{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}f(n)={\frac {1}{2}}f(0)+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{2\sinh(\pi t)}}\,dt,}
which is related to the Lindelöf summation formula [ 2]
∑
k
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m
∞
(
−
1
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k
f
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=
(
−
1
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m
∫
−
∞
∞
f
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m
−
1
/
2
+
i
x
)
d
x
2
cosh
(
π
x
)
.
{\displaystyle \sum _{k=m}^{\infty }(-1)^{k}f(k)=(-1)^{m}\int _{-\infty }^{\infty }f(m-1/2+ix){\frac {dx}{2\cosh(\pi x)}}.}
Let
f
{\displaystyle f}
ℜ
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z
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≥
0
{\displaystyle \Re (z)\geq 0}
f
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0
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=
0
{\displaystyle f(0)=0}
f
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=
O
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|
z
|
k
)
{\displaystyle f(z)=O(|z|^{k})}
arg
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z
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∈
(
−
β
,
β
)
{\displaystyle \operatorname {arg} (z)\in (-\beta ,\beta )}
f
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z
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=
O
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|
z
|
−
1
−
δ
)
{\displaystyle f(z)=O(|z|^{-1-\delta })}
a
=
e
i
β
/
2
{\displaystyle a=e^{i\beta /2}}
residue theorem
∫
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∞
f
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e
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∑
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∞
Res
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=
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∞
f
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.
{\displaystyle \int _{a^{-1}\infty }^{0}+\int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=-2i\pi \sum _{n=0}^{\infty }\operatorname {Res} \left({\frac {f(z)}{e^{-2i\pi z}-1}}\right)=\sum _{n=0}^{\infty }f(n).}
Then
∫
a
−
1
∞
0
f
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z
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e
−
2
i
π
z
−
1
d
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=
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∫
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a
−
1
∞
f
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e
−
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i
π
z
−
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d
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=
∫
0
a
−
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∞
f
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e
2
i
π
z
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d
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+
∫
0
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−
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∞
f
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=
∫
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∞
f
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π
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d
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{\displaystyle {\begin{aligned}\int _{a^{-1}\infty }^{0}{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz&=-\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz\\&=\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{2i\pi z}-1}}\,dz+\int _{0}^{a^{-1}\infty }f(z)\,dz\\&=\int _{0}^{\infty }{\frac {f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\,d(a^{-1}t)+\int _{0}^{\infty }f(t)\,dt.\end{aligned}}}
Using the Cauchy integral theorem for the last one.
∫
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∞
f
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{\displaystyle \int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=\int _{0}^{\infty }{\frac {f(at)}{e^{-2i\pi at}-1}}\,d(at),}
∑
n
=
0
∞
f
(
n
)
=
∫
0
∞
(
f
(
t
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+
a
f
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e
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π
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+
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f
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2
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t
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{\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {a\,f(at)}{e^{-2i\pi at}-1}}+{\frac {a^{-1}f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\right)\,dt.}
This identity stays true by analytic continuation everywhere the integral converges, letting
a
→
i
{\displaystyle a\to i}
∑
n
=
0
∞
f
(
n
)
=
∫
0
∞
(
f
(
t
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+
i
f
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−
i
f
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e
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t
.
{\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {i\,f(it)-i\,f(-it)}{e^{2\pi t}-1}}\right)\,dt.}
The case ƒ (0) ≠ 0 is obtained similarly, replacing
∫
a
−
1
∞
a
∞
f
(
z
)
e
−
2
i
π
z
−
1
d
z
{\textstyle \int _{a^{-1}\infty }^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz}
Abel, N.H. (1823), Solution de quelques problèmes à l'aide d'intégrales définies Butzer, P. L.; Ferreira, P. J. S. G.; Schmeisser, G.; Stens, R. L. (2011), "The summation formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis", Results in Mathematics 59 (3): 359–400, doi :10.1007/s00025-010-0083-8 , ISSN 1422-6383 , MR 2793463 , S2CID 54634413 Olver, Frank William John (1997) [1974], Asymptotics and special functions , AKP Classics, Wellesley, MA: A K Peters Ltd., ISBN 978-1-56881-069-0 MR 1429619 Plana, G.A.A. (1820), "Sur une nouvelle expression analytique des nombres Bernoulliens, propre à exprimer en termes finis la formule générale pour la sommation des suites", Mem. Accad. Sci. Torino , 25 : 403–418 
