= Incomplete beta function
{title2=$B_z(p,q)=\int_0^z u^{p-1}(1-u)^{q-1}\,du$}
For real $p,q>0$ and $0\leq z\leq1$, the displayed integral is the incomplete <beta function>. The regularized version is $I_z(p,q)=B_z(p,q)/B_1(p,q)$, the <cumulative distribution function> of the <beta distribution>. It has endpoint values zero and one and derivative $z^{p-1}(1-z)^{q-1}/B_1(p,q)$. This is a useful explicit <scale function of a one-dimensional diffusion> when the scale density has two power-law endpoint singularities.
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