Searching for an asymptotic to exp[0.5]

[jaydfox]

Have you seen this excellent asymptotic entire series? I can't explain it, other than it matches the "fakefunc" integral

Hey, this reminds me of the generalization I made for the (unscaled) asymptotic binary partition function:

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 (well, any complex number, but the complex versions are less interesting).

We can apply the same generalization to exp(x):



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).

Now, set beta = 1/2 and truncate the negative powers of x:


...and comparing your power series, it easily follows that your power series is asymptotic to exp(x) x^(1/2). Hmm, oh dear, I think you missed a +1 in the Gamma function in your series? (LOL, it's okay, I usually forget the +1 as well.) If you set beta=-1/2, then all is well!