Abel–Plana formula - Alchetron, The Free Social Encyclopedia

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 ( 0 ) + i ∫ 0 ∞ f ( i t ) − f ( − i t ) e 2 π t − 1 d t .

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 .

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 .

References

Abel–Plana formula Wikipedia

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