Trying to Find Closed Form for Beta Integral

Question

$$ \mbox{Consider}\quad\int_{0}^{1/2} v^{-t}\,\left(\, 1 - v\, \right)^{-\,\left(\, t + 1\, \right)} \,\,\,\mathrm{d}v $$ I have verified numerically that the integral converges for $t < 1$, but I can't get a closed form. Is there a way to write this in terms of the Beta function ?.

Answer 1

You could write it in terms of a hypergeometric function:

$$-{\frac {{2}^{-1+t}{\mbox{$_2$F$_1$}(t+1,-t+1;\,2-t;\,1/2)}}{-1+t}}$$