# Unbiased estimator of standard deviation of a normal distribution, using gamma function [duplicate]

According to the Wikipedia article, the following estimator of the standard deviation $$s=\sqrt{\frac{1}{n-1}\sum_{k=1}^n(x_i-\bar{x})^2}$$

for a normal variable, verifies $E[s]=C_4(n) \sigma$, where

$$C_4(n) = \sqrt{\frac{2}{n-1}}\cdot\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n-1}{2})}=1 - \frac{1}{4n} - \frac{7}{32n^2} - \frac{19}{128n^3} + O(n^{-4})$$

Therefore we have an unbiased estimator for the standard deviation of a normal variable,

$$\hat{\sigma}_\text{unbiased} = \frac{1}{C_4(n)} \sqrt{\frac{1}{n-1}\sum_{k=1}^n(x_i-\bar{x})^2}$$

How could you prove this?