--Same post as before, but now I corrected some typos, sorry.
D Asimov: Are there extensions of Bernoulli numbers to real (or complex) index?
Is there one that is generally accepted as the natural way to do it?
It would be nice if there were a natural real-index Bernoulli function that took real values.
I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?
--The generating function for Bernoulli(n)/n! is z/[exp(z)-1]. So we can use Cauchy residue theorem to write Bernoulli(n) = Gamma(n+1)/(2*pi*i) * int z^(1-n) / [exp(z)-1] dz where the contour of integration encircles the origin going counterclockwise and is contained within the strip |Im(z)|<2*pi. Actually, it need not be contained in this strip, if it is shaped right. All we really need is that |Im(z)|<2*pi when Re(z)=0 and that if the contour goes to infinity it must do so in good directions. I would suggest a parabola-like contour which starts at someplace around +i+RealInfinity, stays above the positive real axis, crosses the negative real axis, then staying below the positive real axis goes to someplace near -i+RealInfinity. That way we do not care if the integrand is only defined on a complex z-plane SLIT along the positive real axis. Define the ln function in this slit way and the power function via A^B = exp(B*ln(A)). One concrete choice for the coutour could be the parabola x = y^2 - 1. And then observe that this definition will produce a unique Bernoulli(n) value when n is any complex number, not required to be an integer. Furthermore by reflection symmetry of everything about the real axis we see it always produces real values if n is real because the imaginary part cancels out. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)