Searching for an asymptotic to exp[0.5]

[sheldonison]

How does the difference between fake_exp^[0.5](z) and Kneser_exp^[0.5](z) look like ?
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And how about the difference of fake_exp^[0.5](z) with 2sinh^[0.5](z) ?
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In particular I consider there differences for Re(z) > 0.

Below, is a log_10/log_10 plot of both ratios, from 0.1 to 10^6. The fakeexp asymptotic is accurate to >30 decimal digits at 10^6, using only 55-60 terms. btw, there is also a pretty good asymptotic for sinh^{0.5}, almost as good as the one for exp^{0.5}. Tommy might have been interested in sinh(z)/exp(z) which converges exponentially to '1', much quicker than the fakeexp converges.

\( \exp(x)\sqrt{x} \approx \sum_{n=0}^{\infty} \frac{x^n}{\Gamma(n+0.5)}\;\; \) Have you seen this excellent asymptotic entire series? I can't explain it, other than it matches the "fakefunc" integral