
#include nagmk26.h

$$P\left(T\le t:\nu \right)=\frac{\Gamma \left(\left(\nu +1\right)/2\right)}{\sqrt{\pi \nu}\Gamma \left(\nu /2\right)}\underset{\infty}{\overset{t}{\int}}{\left[1+\frac{{T}^{2}}{\nu}\right]}^{\left(\nu +1\right)/2}dT\text{, \hspace{1em}}\nu \ge 1\text{.}$$ 
(i)  when $\nu <20$, a transformation of the beta distribution, ${P}_{\beta}\left(B\le \beta :a,b\right)$ is used


(ii)  when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970). 