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The Beta function obeysThese identities follow immediately from the beta--gamma identity and the Gamma function recurrence.

With $K=\rho g/(3\mu )$ and formal edge conditions $h(\pm L,t)=0$, the lubrication theory equation is $h_t=K(h^3{h}_x{)}_x$. Its positive separated long-time profile hasWriting $C=B(4/5,1/2)$ in terms of the beta function, the centre height is $h_0^3=10\mu L^2/(\rho g{C}^{2}t)$. The edge region of a draining viscous film determines how the singular outer profile matches the flow over the physical edge.

For $\Re c>\Re b>0$ and a fixed branch away from the cut,Expand $(1-zs{)}^{-a}$ for $|z|<1$ and integrate termwise using the beta function to recover the Gauss hypergeometric function series. Analytic continuation extends the identity in $z$.

In units $G=M=a=1$, velocity integration of $F=C{L}^{p-2}{E}^{(3p+1)/2}$ gives $C=(p+1)\Gamma (2p+2)/[2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)]$. The beta function separates the radial-speed and angular integrals. This constant is positive for every $p>0$.

As $v\to \infty$, $[v(v-1){]}^{-4/\kappa }\sim v^{-8/\kappa }$. ThereforeThus every positive solution is unbounded when κ ≥ 8, including the logarithmically divergent critical case. For $4<\kappa <8$, by contrast,For example the substitution $r=1-1/v$ identifies this normalizing constant as $B(1-4/\kappa ,8/\kappa -1)$, with both beta function parameters positive.