Perhaps everybody has heard of the Stirling's approximation, namely: $$ \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\right)^z $$ Thus (the very basic example): $$ \Gamma\left(\frac{1}{2}z\right) \approx\sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{z}$$ My question is: how does one obtain the lower and upper bound for the Gamma function using the Stirling's approximation? I've heard that $$ \sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{\color{red}{z-1}}<\Gamma\left(\frac{1}{2}z\right)<\sqrt{\frac{4\pi}{z}}\left(\sqrt{\frac{z}{2e}}\right)^{\color{red}{z+1}} $$ is a very "ugly" and unproper way (moreover it doesn't always work). So what's the best way to obtain the lower and upper bound?



If you look here, you will find this very nice inequality $$\sqrt{2\pi}n^{n+\frac 12}e^{-n} \leq n! \leq en^{n+\frac 12}e^{-n}$$ which would give you good approximations of upper and lower bounds for the $\Gamma$ function.

For example $$\Gamma(12.3456)\approx 9.33280\times 10^7$$ while the above formulae give $\approx 9.26452\times 10^7$ and $\approx 1.00468\times 10^8$

$$\Gamma(123.456)\approx 8.85315\times 10^{203}$$ while the above formulae give $\approx 8.84713\times 10^{203}$ and $\approx 9.59416\times 10^{203}$


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