Abel–Plana Formula

In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that

$\sum_^f\left(a+n\right)= \frac+\int_^f\left(x\right)dx+i\int_^\fracdt$

For the case $a=0$ we have
:$\sum_^\infty f(n)=\frac + \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt.$

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. .

An example is provided by the Hurwitz zeta function,
:$\zeta(s,\alpha)= \sum_^\infty \frac = \frac + \frac 1 + 2\int_0^\infty\frac\frac,$
which holds for all $s \in \mathbb$, . Another powerful example is applying the formula to the function $e^n^$: we obtain

$\Gamma(x+1)=\operatorname_\left(e^\right)+\theta(x)$ where $\Gamma(x)$ is the gamma function, $\operatorname_\left(z\right)$ is the polylogarithm and $\theta(x)=\int_^\frac\sin\left(\frac-t\right)dt$.

Abel also gave the following variation for alternating sums:
:$\sum_^\infty (-1)^nf(n)= \frac f(0)+i \int_0^\infty \frac \, dt,$
which is related to the Lindelöf summation formula

: $\sum_^\infty (-1)^kf(k)=(-1)^m\int_^\infty f(m-1/2+ix)\frac.$

Proof

Let $f$ be holomorphic on $\Re(z) \ge 0$, such that $f(0) = 0$, $f(z) = O(, z, ^k)$ and for $\operatorname(z)\in (-\beta,\beta)$, $f(z) = O(, z, ^)$. Taking $a=e^$ with the residue theorem
$\int_^0 + \int_0^ \frac \, dz = -2i\pi \sum_^\infty \operatorname\left(\frac\right)=\sum_^\infty f(n).$

Then
$\begin \int_^0 \frac \, dz&=-\int_0^ \frac \, dz \\ &=\int_0^\frac \, dz+\int_0^ f(z) \, dz\\ &= \int_0^\infty \frac \, d(a^t)+\int_0^\infty f(t) \, dt. \end$

Using the Cauchy integral theorem for the last one. $\int_0^ \frac \, dz = \int_0^\infty \frac \, d(at),$ thus obtaining
$\sum_^\infty f(n)=\int_0^\infty \left(f(t)+\frac + \frac\right) \, dt.$

This identity stays true by analytic continuation everywhere the integral converges, letting $a\to i$ we obtain the Abel–Plana formula
$\sum_^\infty f(n)=\int_0^\infty \left(f(t)+\frac\right) \, dt.$

The case ''ƒ''(0) ≠ 0 is obtained similarly, replacing $\int_^ \frac \, dz$ by two integrals following the same curves with a small indentation on the left and right of 0.

See also

* Euler–Maclaurin summation formula
* Euler–Boole summation
* Ramanujan summation

References

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External links

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{{DEFAULTSORT:Abel-Plana formula
Summability methods