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For $\mathrm{Re}p,\mathrm{Re}q>0$, Euler's beta function is

It follows by making the logarithm of the quotient entire and periodic; growth forces its periodic remainder to be constant.

The digamma function is the logarithmic derivativeLogarithmically differentiating the Weierstrass product gives

The reciprocal gamma function has the entire-product representation

This is a particular application of the Weierstrass factorization theorem.

For $\mathrm{Re}s>1$, expansion of $(e^x-1{)}^{-1}$ as a geometric series and termwise integration give

The Gamma function obeys the recurrence relationInitially, for $\mathrm{Re}z>0$, integration by parts in the Gamma integral gives the short proofThe boundary term vanishes at zero because $|t^z|=t^{\mathrm{Re}z}\to 0$, and at infinity because of exponential decay. Together with $\Gamma (1)={\int }_0^{\infty }{e}^{-t}dt=1$, the Gamma function recurrence immediately gives $\Gamma (n+1)=n!$ for every nonnegative integer $n$.^[ref]

Read backwards, the Gamma function recurrence defineswhere $m$ is a positive integer. These overlapping formulas agree by the Gamma function recurrence, giving the meromorphic continuation. Since $\Gamma (1)=1$, they give a simple pole at zero and then at every negative integer; Gamma function residue at a nonpositive integer records the residues. The recurrence relation therefore holds throughout the continuation as an identity of meromorphic functions.

The discriminant square class detects whether the transitive cubic Galois group lies in the alternating group.

The polynomial $X^3-X-1$ is irreducible over $\mathbb{Q}$ and has discriminant $-23$, so its splitting field has Galois group $S_3$. Its unique quadratic subfield is the fixed field of $A_3$, namely $\mathbb{Q}(\sqrt{-23})$.

Suppose a group of field automorphisms acts transitively on elements $y_i$, and each $y_i$ has a unique $p$th root $x_i$ in a stable extension. If every automorphism extends, it sends $x_i$ to the unique root above its image of $y_i$, so the extended group acts transitively on the $x_i$.

Put $\alpha =\sqrt[4]{7}$ and $L=\mathbb{Q}(\alpha ,i)$. The dihedral group $D_8=⟨r,s:r^4=s^2=1,srs=r^{-1}⟩$ acts byThe ten subgroups and their fixed fields areThe normal subgroups are $D_8$, $⟨r⟩$, the two displayed Klein four-groups, $⟨r^2⟩$, and $\{1\}$.

Every subgroup of an abelian group is normal. Hence every intermediate field of a finite abelian Galois extension is itself Galois over the base field.

For a finite Galois extension $L/K$ with group $G$, an intermediate extension $E/K$ is normal exactly when its corresponding subgroup $\mathrm{Gal}(L/E)$ is a normal subgroup of $G$.

The disk--halo degeneracy is the ability of different stellar mass-to-light ratios and dark-halo profiles to reproduce nearly the same total galaxy rotation curve.

A thin exponential galactic disk has surface density $\Sigma (R)={\Sigma }_0{e}^{-R/R_d}$ and total mass $2\pi {\Sigma }_0{R}_d^2$.

A singular isothermal sphere with one-dimensional velocity dispersion $\sigma$ has $\rho (r)={\sigma }^2/(2\pi G{r}^2)$, enclosed mass $M(<r)=2{\sigma }^{2}r/G$, and constant circular speed $\sqrt{2}\sigma$.

Stellar dynamics studies the collective gravitational motion of stars in galaxies and star clusters.

The tidal radius bounds material that remains gravitationally attached to a satellite in the differential gravitational field of its host.

Chandrasekhar dynamical friction is the drag on a massive body produced by its gravitational wake in a background of lighter particles. Its magnitude is proportional to the body's mass and the background density.

A European contingent claim pays a specified function of market variables at one fixed maturity. Under an equivalent martingale measure, an attainable claim is priced by the discounted conditional expectation of its payoff.