Integral Representation and Continuation
The integral
has as special values b(2n) = B2n for n > 0. The integral might be considered as a continuation of the Bernoulli numbers to the complex plane and this was indeed suggested by Peter Luschny in 2004.
For example b(3) = (3/2)ζ(3)Π−3Ι and b(5) = −(15/2) ζ(5) Π −5Ι. Here ζ(n) denotes the Riemann zeta function and Ι the imaginary unit. It is remarkable that already Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated
Euler's values are unsigned and real, but obviously his aim was to find a meaningful way to define the Bernoulli numbers at the odd integers n > 1.
Read more about this topic: Bernoulli Number
Famous quotes containing the words integral and/or continuation:
“Make the most of your regrets; never smother your sorrow, but tend and cherish it till it come to have a separate and integral interest. To regret deeply is to live afresh.”
—Henry David Thoreau (1817–1862)
“Institutional psychiatry is a continuation of the Inquisition. All that has really changed is the vocabulary and the social style. The vocabulary conforms to the intellectual expectations of our age: it is a pseudo-medical jargon that parodies the concepts of science. The social style conforms to the political expectations of our age: it is a pseudo-liberal social movement that parodies the ideals of freedom and rationality.”
—Thomas Szasz (b. 1920)