Actually, assuming wikipedia's statement of Van Vleck's convergence theorem is correct, we see that a continued fraction y/(a + y/(b + y/(c + ... ))) if all the a,b,c... are positive, will obey 1) its even convergents converge 2) its odd ones do too 3) the two limits are the same if a+b+c+... = infinity all PROVIDED |arg(y)|<pi-epsilon. So in particular the continued fraction I devised for a Stirling-like gamma-function approximation, should converge everywhere in any complex z-halfplane Re(z)>epsilon>0 PROVIDED it is really true that all its coefficients keep on being positive and don't shrink too quickly. This positivity is a remarkable thing. I am not seeing a proof of it, there must be some deep reason for it and perhaps some quite general underlying theorems lurk saying that if you do certain things to continued fractions, their coefficients stay positive. (The other remarkable thing, that the series only involves odd powers, can be proven using methods in my paper that date back about 100 years; that's comparatively trivial.) Note that the odd and even convergents will provide bracketing bounds on the true limit (if z>0 is real) so you can easily tell when to stop working, you have enough precision. I would suggest to Munafo that he based on experiment compute a table saying how many terms of this CF are required for different z to get different levels of precision. Then we can assess how quickly it is converging and whether it is competitive with other gamma approximations.