# Find the approximation with Gamma function

Let $n$ postive integer,Assmue that $\Gamma(x)$ is Gamma function $$\dfrac{\Gamma\left(\dfrac{1}{2}+\dfrac{1}{2n}\right)}{\Gamma\left(1+\dfrac{1}{2n}\right)}=A+\dfrac{B}{n}+\dfrac{C}{n^2}+\dfrac{D}{n^3}+\cdots$$

use $$\dfrac{d}{dz}\left(\ln{\Gamma\left(\dfrac{1}{2}+z\right)}-\ln{\Gamma\left(1+z\right)}\right)=\varepsilon(z+1/2)-\varepsilon(z+1)$$ where $\varepsilon(z)$ is Digamma function Now I have prove $$A=\sqrt{\pi},B=-\sqrt{\pi}\log{2}$$ But can't find $C,D$

Considering $$A(z)=\log(Y(z))=\log \left(\Gamma \left(\frac{1}{2}+z\right)\right)-\log (\Gamma (z+1))$$ you are looking for the Taylor expansion built around $z=0$. This means $$A(z)= A(0)+\frac{A'(0)} {1!}z+\frac{A''(0)} {2!}z^2+\frac{A'''(0)} {3!}z^3+\cdots$$

You are given for the derivatives $$A'(z)=\psi ^{(0)}\left(z+\frac{1}{2}\right)-\psi ^{(0)}(z+1)$$ which generalizes to $$A''(z)=\psi ^{(1)}\left(z+\frac{1}{2}\right)-\psi ^{(1)}(z+1)$$ $$A'''(z)=\psi ^{(2)}\left(z+\frac{1}{2}\right)-\psi ^{(2)}(z+1)$$

which, now, need to be simplified using $z=0$.

So $$A(0)=\frac{\log (\pi )}{2}\qquad A'(0)=-\log (4)\qquad A''(0)=\frac{\pi ^2}{3}\qquad A'''(0)=-12 \zeta (3)$$ This makes $$A(z)=\frac{\log (\pi )}{2}-\log (4)z+\frac{\pi ^2 }{6}z^2-2 \zeta (3)z^3+O\left(z^4\right)$$ Now, Taylor again $$Y(z)=e^{A(z)}=\sqrt{\pi }-\sqrt{\pi } \log (4)z+\frac{1}{6} \sqrt{\pi } \left(\pi ^2+3 \log ^2(4)\right)z^2-\frac{1}{6} \left(\sqrt{\pi } \left(12 \zeta (3)+\log ^3(4)+\pi ^2 \log (4)\right)\right)z^3 +O\left(z^4\right)$$ Now, to finish, replace $z$ by $\frac 1 {2n}$.

Just to see how accurate is the expansion, $$\left( \begin{array}{ccc} n & \text{exact} & \text{approximation}\\ 5 & 1.565345082 & 1.563836840 \\ 10 & 1.660110105 & 1.660007284 \\ 15 & 1.695365291 & 1.695344347 \\ 20 & 1.713776688 & 1.713769957 \end{array} \right)$$

Doing a Google search for "gamma function asymptotics" leads to a number of relevant links including this:

http://dlmf.nist.gov/5.11

This, in turn, has a number of references. This one is free:

T. Burić and N. Elezović (2011) Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math. 235 (11), pp. 3315–3331.

It is available at

http://www.sciencedirect.com/science/article/pii/S0377042711000562