Evaluating the ratio of beta functions I came across a question which asks for the value of $\alpha$ in the result to the ratio of two $\mathrm{B}$ functions:
$$\frac{\mathrm{B}(m, \frac{1}{2})}{\mathrm{B}(m, m)}=2^{\alpha}$$
I know the results for integer values of $m$, but the question demands that $m>0$ be any real number.
I also tried changing the functions to $\Gamma$ ones, but that didn't lead anywhere meaningful.
Does anyone know how to approach this?
P.S. It seems to be an easy question: the marks to be awarded against it is unity in modulus. 

For any value of $m$
$$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=2^{2m-1}$$
If you use the gamma function
$$B(m,n)=\frac{\Gamma (m) \Gamma (n)}{\Gamma (m+n)}$$
$$\frac{{B}(m, \frac{1}{2})}{{B}(m, m)}=\frac {\sqrt{\pi }\frac{ \Gamma (m)}{\Gamma \left(m+\frac{1}{2}\right)} } {\frac{\Gamma (m)^2}{\Gamma (2 m)} }=\sqrt{\pi }\frac{ \Gamma (2 m)}{\Gamma (m) \Gamma \left(m+\frac{1}{2}\right)}$$Use Stirling approximation up to any order of your choice, continue with Taylor series to prove it.
