09/09/2014, 07:43 PM

(08/08/2014, 12:55 AM)jaydfox Wrote: Deriving the continuous function is actually pretty straightforward.

(...)

In general:

By induction:

Then, as I posted previously:

I'm kind of disappointed I didn't see this earlier. I didn't even arrive at it by "first principles". Rather, I was playing around with creating a function that obeys the functional equation f'(x) = f(x/2), but which oscillates in a similar manner to the way the binary partition function oscillates. It's actually pretty cool, especially the code I came up with. Perhaps I'll share in a later post.

At any rate, it dawned on me that the oscillating version of the function behaves like a Laurent series, with positive powers of x creating oscillations far from the origin (with a periodicity of about two zeroes per doubling of scale), and negative powers of x creating oscillations near the origin (with a periodicity of about one zero per doubling of scale).

And as I was trying to formalize it, I realized something that I'd missed earlier, which should have been obvious:

Treating Gamma(k) at negative integers as infinity, and the reciprocal of such as zero, we can take the limit from negative to positive infinity. And we can replace k with (k+b), where b is zero in the original solution, but can now be treated as any real (well, any complex number, but the complex versions are less interesting).

So, without further ado, here are some graphs comparing f_{1/2} to f_0:

~ Jay Daniel Fox