Where I suppose for x > 0 real we define HarmonicNumber[x] := Integral_{0,1} (1 - u^x) / (1 - x) dx ??? (Which is 1 + 1/2 + 1/3 + ... + 1/n for n in Z+.) --Dan
On Dec 23, 2014, at 3:03 PM, Olivier Gerard <olivier.gerard@gmail.com> wrote:
On Tue, Dec 23, 2014 at 1:54 PM, Daniel Asimov <asimov@msri.org> wrote:
Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
Very delicate hypothesis. You would have to prove that there is a continuous family of geometrical objects, to whom volume and surface make sense, etc. and according precisely to these formulae.
upsilon := 0.256946404860576780132838388690769236619+
Is it known to be irrational or transcendental? Or related to other
numbers,
like Euler gamma, whose number-theoretic properties are unknown?
Simplifying the expression for the derivative of Vmax relative to d, you get that d_Vmax is such that
EulerGamma + Log[Pi] == HarmonicNumber[d_Vmax / 2]
in Mathematica notation, or equivalently
Log[Pi] == PolyGamma[0, d_Amax / 2]