09/15/2014, 03:53 AM
(This post was last modified: 09/15/2014, 04:33 AM by sheldonison.)

(09/13/2014, 11:49 PM)jaydfox Wrote: ...

Treating Gamma(k) at negative non-positive 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...

Notice that I put "approximately equal". I haven't checked, but I assume it's exactly equal, but only in the sense that it should satisfy the functional equation exp'(x) = exp(x).

So we also have a fakeexp(z) function, and I especially like the simplicity of f'(x)=f(x) in the non-converging limit to explain why f(x)~exp(x). Here, going back to k=0.5, for simplicity.... We have to put some bounds on k, since the infinite Laurent series does not converge anywhere.

As gets arbitrarily large, the error term becomes more and more insignificant, relative to the exp(z) term, and the number of terms you can include also increases, until the starts growing in magnitude as n gets bigger negative ....

- Sheldon