Abel–Plana formula

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

For the case we have

It holds for functions ƒ that are holomorphic in the region Re(z) Ã¢Â‰Â¥ 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,

which holds for all , . Another powerful example is applying the formula to the function : we obtain

where is the gamma function, is the polylogarithm and .

Abel also gave the following variation for alternating sums:

which is related to the Lindelöf summation formula

Proof

Let be holomorphic on , such that , and for , . Taking with the residue theorem

Then

Using the Cauchy integral theorem for the last one. thus obtaining

This identity stays true by analytic continuation everywhere the integral converges, letting we obtain the Abel–Plana formula

The case ƒ(0) ≠ 0 is obtained similarly, replacing 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

External links