Evaluating the Gamma function in Bessel's equation

Question

I'm trying to evaluate the Gamma function in Bessel's equation for $v=1/4$.

What is $\Gamma(m+5/4)$, or when solving, what is $\Gamma(1/4)$?

Answer 1

$$\Gamma\left(m+\frac{1}{4}\right)=\Gamma\left(\frac{1}{4}\right)\prod _{k=1}^m\left(k-\frac{3}{4}\right)$$ $\Gamma\left(\frac{1}{4}\right)$ is a remarkable constant. $$\Gamma\left(\frac{1}{4}\right)\simeq 3.62560990822190831...$$ In the book "An Atlas of Functions",J.Spannier, K.B.Oldham, this constant is noted U and called the "ubiquitous constant". It appears throughout many relationships involving special functions. One most notable is : $$\Gamma\left(\frac{1}{4}\right)=2\sqrt{\sqrt{\pi}\text{K}\left(1/2\right)}$$ where K$(x)$ is the complete elliptic integral of the first kind.

An efficient iterative process to evaluate $\Gamma\left(\frac{1}{4}\right)$ is : $$\Gamma\left(\frac{1}{4}\right)=\text{Arithmetic-Geometric Mean of }1 \text{ and }\frac{1}{\sqrt{2}}$$

The arithmetic-geometric mean of two numbers $A_0\:,\:B_0$ is approached by repeating : $A_{k+1}=(A_k+B_k)/2$ and $B_{k+1}=\sqrt{A_kB_k}$.

Answer 2

Some properties of that value are gathered in the associated wikipedia article.