Abel–Plana formula explained

In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that ^[1]

\sum ^{inf ty} _{n =0} f\left(a + n\right) = \int ^{inf ty} _{a} f\left(x\right)dx + (f\left(a\right))/(2) + \int ^{inf ty} _{0} (f\left(a - ix\right) - f\left(a + ix\right))/(i\left(e ^{2\pi} -1 \right)) dx

For the case

a =0 we have\sum ^{inf ty} _{n =0} f(n) = (1)/(2) f( 0 ) + \int ^{inf ty} _{0} f(x) dx + i \int ^{inf ty} _{0} (f(i t) - f( - i t))/(e ^{2\pi} -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. .

An example is provided by the Hurwitz zeta function,

\zeta(s,\alpha) = \sum ^{inf ty} _{n =0} (1)/((n + \alpha) ^{s}) = (\alpha ^{1-s})/(s -1) + (1 { 2 \alpha)/(^{s}}) + 2 \int ^{inf ty (\sin\left(s \arctan (t)/(\alpha\right)))/((\alpha ^{2+ t} ^{2)} ^{(s)/(2)})} _{0} (dt)/(e ^{2\pi} -1) , which holds for all s \in C , . Another powerful example is applying the formula to the function e ^{-n} n ^{x} : we obtain \Gamma(x +1 ) = \operatorname{Li} _{-x} \left(e ^{-1} \right) + \theta(x) where \Gamma(x) is the gamma function, \operatorname{Li} _{s} \left(z\right) is the polylogarithm and \theta(x) = \int ^{inf ty} _{0} (2 t ^{x})/(\sin\left( e ^{2\pi} -1) (\pi x)/(2) - t\right)dt .

Abel also gave the following variation for alternating sums:

\sum ^{inf ty} _{n =0} ( -1 ) ^{nf(n)=} (1)/(2) f( 0 ) + i \int ^{inf ty} _{0} (f(i t) - f( - i t))/(2 \sinh(\pi t)) dt, which is related to the Lindelöf summation formula ^[2] \sum ^{inf ty} _{k = m} ( -1 ) ^{kf(k)=(-1)} f(m -1 / 2+ ix) ^{inf ty} _{-inf ty} (dx)/(2 \cosh(\pi x)) .

Proof

Let

f be holomorphic on \Re(z) \ge 0 , such that f( 0 ) = 0 , f(z) = O(|z| ^{k)} and for \operatorname{arg}(z)\in ( - \beta,\beta) , f(z) = O(|z| ^{-1-\delta} ) . Taking a = e ^{i} with the residue theorem $$\backslash int\_^0\; +\; \backslash int\_0^\; \backslash frac\; \backslash ,\; dz\; =\; -2i\backslash pi\; \backslash sum\_^\backslash infty\; \backslash operatorname\backslash left(\backslash frac\backslash right)=\backslash sum\_^\backslash infty\; f(n).$$

Then

Using the Cauchy integral theorem for the last one. $$\backslash int\_0^\; \backslash frac\; \backslash ,\; dz\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; d(at),$$ thus obtaining$$\backslash sum\_^\backslash infty\; f(n)=\backslash int\_0^\backslash infty\; \backslash left(f(t)+\backslash frac\; +\; \backslash frac\backslash right)\; \backslash ,\; dt.$$

This identity stays true by analytic continuation everywhere the integral converges, letting

a\to i we obtain the Abel–Plana formula$$\backslash sum\_^\backslash infty\; f(n)=\backslash int\_0^\backslash infty\; \backslash left(f(t)+\backslash frac\backslash right)\; \backslash ,\; dt.$$

The case ƒ(0) ≠ 0 is obtained similarly, replacing $\backslash int\_^\; \backslash frac\; \backslash ,\; dz$ 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

Notes and References

1. Hermite. C.. Extrait de quelques lettres de M. Ch. Hermite à M. S. Píncherle. Annali di Matematica Pura ed Applicata . Serie III. 5. 1901. 57–72.
2. Web site: Summation Formulas of Euler-Maclaurin and Abel-Plana: Old and New Results and Applications.