Bernoulli Number - Integral Representation and Continuation

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

\begin{align} p &= \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\text{etc.}\ \right) = 0.0581522\ldots \\ q &= \frac{15}{2\pi^{5}}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\text{etc.}\ \right) = 0.0254132\ldots \end{align}

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

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