Proof for approximation $\Gamma(x+\frac{1}{2})\approx\sqrt{x}\,\Gamma(x)$ I am working on finding that $\Gamma(x+\frac{1}{2})$ can be approximated with $\sqrt{x}\Gamma(x)$. I have tried many approximations, e.g., Stirling, Taylor series, etc, but I cannot find an appropriate solution. Can anyone give me any guidelines? Thanks!
Update: I forgot to mention that they almost match in Matlab simulations. 
e.g., plot([1:100],sqrt([1:100]),'r');hold on;plot([1:100],gamma([1:100]+1/2)./gamma([1:100]),'o')
 A: A particular case of Gautschi's inequality gives
$$\forall x>0,\qquad \sqrt{x}<\frac{\Gamma(x+1)}{\Gamma\left(x+\frac{1}{2}\right)}<\sqrt{x+1} \tag{1}$$
hence
$$\frac{x}{\sqrt{x+1}}<\frac{\Gamma\left(x+\frac{1}{2}\right)}{\Gamma(x)}<\sqrt{x}. \tag{2}$$
Gautschi's inequality is a quite straightforward consequence of the log-convexity of the $\Gamma$ function.

A more accurate approximation comes from Stirling's inequality
$$ \log\Gamma(x) = \left(x-\frac{1}{2}\right)\log x-x+\frac{1}{2}\log(2\pi)+\frac{O(1)}{x}\tag{3} $$
leading to:
$$ \log\frac{\Gamma\left(x+\frac{1}{2}\right)}{\Gamma\left(x\right)} -\left(\frac{1}{2}\log x-\frac{1}{8x}\right) = \frac{O(1)}{x^3}\tag{4}$$

Essentially the same accuracy can also be recovered from the fact that

If $\{a_n\}_{n\geq 1}$ is defined through $$
 a_n=\binom{2n}{n}\frac{\sqrt{\pi\left(n+\frac{1}{4}\right)}}{4^n} $$
   $\{a_n\}_{n\geq 1}$ is an increasing sequence converging towards $1$.

This fact can be proved through Weierstrass products and creative telescoping, too.
The Legendre duplication formula then gives
$$ \frac{\Gamma\left(x+\frac{1}{2}\right)}{\Gamma(x)}\sim \frac{x}{\sqrt{x+\frac{1}{4}}}. \tag{5}$$
A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
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Stirling Asymptotic Expansion '$+$' Duplication Formula:

\begin{align}
\ln\pars{\Gamma\pars{z + {1 \over 2}}} & \sim
z\ln\pars{z} - z + {1 \over 2}\,\ln\pars{2\pi} - {1 \over 24z}
+
{7 \over 2880z^{3}} - {31 \over 40320z^{5}}  + \mrm{O}\pars{1 \over z^{7}}
\\[5mm]
\ln\pars{\Gamma\pars{z}}
& \sim
\pars{z - \color{#f00}{1 \over 2}}\ln z - z + {1 \over 2}\ln\pars{2\pi} + \frac{1}{12z}
-
\frac{1}{360z^{3}} + {1 \over 1260z^{5}} + \mrm{O}\pars{1 \over z^{7}}
\end{align}

$$
\ln\pars{\Gamma\pars{z + 1/2} \over \root{z}\Gamma\pars{z}} \sim
-\,{1 \over 8z} + {1 \over 192z^{3}} + \mrm{O}\pars{1 \over z^{5}}\quad
\mbox{as}\ \verts{z} \to \infty
$$
