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
∑
n
=
0
∞
f
(
n
)
=
∫
0
∞
f
(
x
)
d
x
+
1
2
f
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0
)
+
i
∫
0
∞
f
(
i
t
)
−
f
(
−
i
t
)
e
2
π
t
−
1
d
t
.
{\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }f(x)\,dx+{\frac {1}{2}}f(0)+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.}
It holds for functions f 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 |f | 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
(
s
,
α
)
=
∑
n
=
0
∞
1
(
n
+
α
)
s
=
α
1
−
s
s
−
1
+
1
2
α
s
+
2
∫
0
∞
sin
(
s
arctan
t
α
)
(
α
2
+
t
2
)
s
2
d
t
e
2
π
t
−
1
,
{\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
Abel also gave the following variation for alternating sums:
∑
n
=
0
∞
(
−
1
)
n
f
(
n
)
=
1
2
f
(
0
)
+
i
∫
0
∞
f
(
i
t
)
−
f
(
−
i
t
)
2
sinh
(
π
t
)
d
t
.
{\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.}
Proof [ ] Let
f
{\displaystyle f}
ℜ
(
z
)
≥
0
{\displaystyle \Re (z)\geq 0}
f
(
0
)
=
0
{\displaystyle f(0)=0}
f
(
z
)
=
O
(
|
z
|
k
)
{\displaystyle f(z)=O(|z|^{k})}
arg
(
z
)
∈
(
−
β
,
β
)
{\displaystyle {\text{arg}}(z)\in (-\beta ,\beta )}
f
(
z
)
=
O
(
|
z
|
−
1
−
δ
)
{\displaystyle f(z)=O(|z|^{-1-\delta })}
a
=
e
i
β
/
2
{\displaystyle a=e^{i\beta /2}}
residue theorem
∫
a
−
1
∞
0
+
∫
0
a
∞
f
(
z
)
e
−
2
i
π
z
−
1
d
z
=
−
2
i
π
∑
n
=
0
∞
R
e
s
(
f
(
z
)
e
−
2
i
π
z
−
1
)
=
∑
n
=
0
∞
f
(
n
)
.
{\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 }\mathrm {Res} \left({\frac {f(z)}{e^{-2i\pi z}-1}}\right)=\sum _{n=0}^{\infty }f(n).}
Then
∫
a
−
1
∞
0
f
(
z
)
e
−
2
i
π
z
−
1
d
z
=
−
∫
0
a
−
1
∞
f
(
z
)
e
−
2
i
π
z
−
1
d
z
=
∫
0
a
−
1
∞
f
(
z
)
e
2
i
π
z
−
1
d
z
+
∫
0
a
−
1
∞
f
(
z
)
d
z
=
∫
0
∞
f
(
a
−
1
t
)
e
2
i
π
a
−
1
t
−
1
d
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a
−
1
t
)
+
∫
0
∞
f
(
t
)
d
t
.
{\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.
∫
0
a
∞
f
(
z
)
e
−
2
i
π
z
−
1
d
z
=
∫
0
∞
f
(
a
t
)
e
−
2
i
π
a
t
−
1
d
(
a
t
)
,
{\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),}
thus obtaining
∑
n
=
0
∞
f
(
n
)
=
∫
0
∞
(
f
(
t
)
+
a
f
(
a
t
)
e
−
2
i
π
a
t
−
1
+
a
−
1
f
(
a
−
1
t
)
e
2
i
π
a
−
1
t
−
1
)
d
t
.
{\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
)
+
i
f
(
i
t
)
−
i
f
(
−
i
t
)
e
2
π
t
−
1
)
d
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 f (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}
0 .
See also [ ] References [ ] 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 External links [ ] 
