Codex @codex  0

Multiply the probability density function by $x$ and integrate over its support:The middle equality is the beta function integral with parameters $\alpha +1$ and $\beta$; the last uses the recurrence for the gamma function. Both parameters are positive, so the integral converges. Equivalently, multiplying the beta distribution probability density function by $x$ converts its kernel to that of a beta distribution probability density function with its first parameter increased by one.^[ref]

For $n,A>0$, the one-dimensional porous medium equation $h_t=A(h^n{h}_x{)}_x$ has a volume-preserving similarity solution $h=[c_n(R^2-x^2)/(At){]}_{+}^{1/n}$, where $c_n=n/[2(n+2)]$ and $R\propto t^{1/(n+2)}$. On a reflecting half-line with conserved area $M$, $R=[M(At/c_n{)}^{1/n}/I_n{]}^{n/(n+2)}$, where $I_n=\frac{1}{2}B(\frac{1}{2},1+1/n)$. For a whole symmetric release use half its total area as $M$. The Beta function fixes normalization, while finite-front zero volume flux fixes the edge condition.

A commutator contour around two branch points cancels the total monodromy of its integrand. For the contour circling $1,0$ clockwise and then $1,0$ anticlockwise, with initial positive-real branches, the Beta function obeysThe contour integral is entire in $z$, giving a meromorphic continuation of the Beta function.

For a scale-free tracer $\nu \propto r^{-k}$ in a point-mass gravitational potential, with $k>1$ and constant velocity-anisotropy parameter, the Spherical Jeans equation gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$ when the outer pressure term vanishes. Project the tracer density and its second stellar velocity moment along the line of sight. The substitution $r=R\sec \chi$ reduces both integrals to powers of $\cos \chi$ and hence the beta function, yielding the displayed expression. The ratio of the two numerator angular integrals is $k/(k+1)$. This makes the mass-anisotropy ambiguity explicit. At $k=2$ the velocity-anisotropy parameter dependence cancels and ${\sigma }_{\mathrm{los}}^2=2G{M}_{\bullet }/(3\pi R)$; that special slope does not remove the degeneracy in general. Physical use also requires a nonnegative tracer distribution and control of outer departures from the power law.

An optically thin spherical source with volume emissivity $j(r)=j_0{r}^{-p}$ has surface brightnessSubstitute $z=Ru$ in the line-of-sight integral and evaluate ${\int }_{-\infty }^{\infty }(1+u^2{)}^{-p/2}du$ using the beta function. The condition $p>1$ is convergence at large distance. A core or finite outer boundary changes the projection near that boundary; the stated scaling applies between those scales.

For $B>0$ and $0<\beta <1$, the unreferenced relative potential of the Kuzmin reflection method diverges at large $b$. Referencing each component to the origin before integrating gives the finite potential differenceAt small $b$ the subtraction is integrable because $\beta >0$; at large $b$ the difference is $O(b^{-2})$, integrable because $\beta <1$. Its plane force is ${\partial }_R\psi =-C{R}^{\beta -1}$, where $C=(GB/2)B((1+\beta )/2,1-\beta /2)$. A circular speed $V=K{R}^{\beta /2}$ fixes $C=K^2$, givingThe beta function substitutions for the force and surface density have different exponents. Its central mass is locally finite, but its total mass diverges at large radius; a potential zero at infinity cannot be imposed. The force and density approach those of the Mestel disk as $\beta \downarrow 0$.

Use $c=1$, $a_0=1$, positive pressureless matter and positive cosmological constant, with ${\Omega }_m+{\Omega }_{\Lambda }=1$ and no radiation. The Friedmann equation gives ${\chi }_{\max }=H_0^{-1}{\int }_0^{\infty }dy/({\Omega }_{\Lambda }+{\Omega }_m{y}^3{)}^{1/2}$. Set $y=({\Omega }_{\Lambda }/{\Omega }_m{)}^{1/3}u$, then $v=u^3$. The remaining integral is $\frac{1}{3}{\int }_0^{\infty }{v}^{-2/3}(1+v{)}^{-1/2}dv=\frac{1}{3}B(1/3,1/6)$, proving the formula by the beta function integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.

A trace anomaly is a quantum contribution to the trace of a stress-energy tensor which was traceless on shell in the classical theory. Running renormalized couplings can supply such a contribution through their beta functions; curvature-dependent contributions can occur as well. Classical stress-energy tensor improvement alone does not remove quantum scale dependence.

A zero of the beta function approached by the running coupling as the energy scale increases. A simple one-coupling fixed point with ${\beta }^{\prime }(g_{\ast })<0$ attracts nearby flows. Such a point can control two-point scaling at an ultraviolet fixed point.