The beta-gamma identity gives $\Gamma (z/2{)}^2=B(z/2,z/2)\Gamma (z)$ for $\mathrm{Re}z>0$. A zero of $\Gamma (z)$ would therefore force zeros at every $z/2^m$. But $w\Gamma (w)=\Gamma (1+w)\to 1$ as $w\to 0$ in the right half-plane, a contradiction. The Gamma function recurrence then extends nonvanishing to every point where the meromorphic gamma function is finite. Nonpositive integers are poles, not zeros.

For positive real $a$, differentiating the gamma function integral under the integral sign gives the displayed formula; successive derivatives insert higher powers of $\log t$. Dominated differentiation on compact positive-$a$ intervals is justified by exponential decay at infinity and integrability of $t^{a-1}|\log t{|}^n$ at zero. At $a=1$, these derivatives give moments used in logarithmic endpoint asymptotic expansions.

For a Galton-Watson process, replacing an individual's reproduction by a one-descendant uneventful step with probability $q<1$ changes its offspring probability generating function to the displayed mixture. Its fixed-point equation reduces to $G(z)=z$, so the smallest fixed point and hence the extinction probability of a branching process are unchanged. The mean is $q+(1-q)G^{\prime }(1)$, which also preserves the subcritical/critical/supercritical classification. The excluded case $q=1$ never reproduces or dies and need not share the extinction probability.

Let $R$ be an integrally closed domain and $f\in R[X]$ monic, with discriminant outside a prime ideal $P$. The roots stay distinct after reduction. In the splitting algebra choose a prime above $P$. Its decomposition group maps onto the Galois group of the residue splitting field; its inertia group fixes every reduced root and is trivial because reduction bijects the distinct root sets. Thus the residue Galois group embeds in the generic group as a permutation group. For finite residue fields, irreducible factor degrees give the Frobenius cycle type.

For an abelian variety $A$ over a field $K$, its Weil–Châtelet group is the Galois cohomology group $H^1(K,A(K^{\mathrm{sep}}))$, classifying torsors under $A$. Such an algebraic-group torsor becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an elliptic curve, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the Tate–Shafarevich group.

Every red-blue finite coloring of the pairs of positive integers has either a blue complete arithmetic progression of any prescribed finite length $m$, or two disjoint $m$-term arithmetic progressions with all cross-pairs red. To see the geometry, assign $(a,d)$ a chosen red pair of indices $(r,s)$ in its progression, assuming the first alternative fails. Apply the Gallai theorem for an integer lattice to the fixed finite patternIn a monochromatic copy $(a_0,d_0)+qF$ with common label $(r,s)$, the selected red pairs are preciselyThe copy lies in the positive quadrant, so $d_0>q(m-1)$. Consequently the two displayed arithmetic progressions are disjoint. Translating the finite pattern before applying the lattice theorem permits its initially negative coordinates.

For a finite pattern $\{v_1,\dots ,v_k\}$ in a real vector space, pull a finite colouring back to words through the displayed sum map. A combinatorial line with active coordinates J maps to $a+|J|\{v_1,\dots ,v_k\}$. The Hales-Jewett theorem therefore forces a monochromatic homothetic copy. Integer patterns remain in the integer lattice; choosing b sufficiently large gives the positive-lattice version.

Every colouring of the positive integers, even with infinitely many colours, has an arithmetic progression of any prescribed finite length on which the colouring is constant or injective. Encode a progression by the equivalence relation of equality of its colours. This gives a finite colouring of its starting point and step. A Gallai theorem for an integer lattice copy of a suitable two-dimensional pattern makes either all indices inequivalent, or one equal-colour pair range independently over two whole progressions. In the second case, one of those progressions is monochromatic.

A survey window function describes the finite spatial coverage and weights of a galaxy survey. Multiplication by this window in position space mixes Fourier modes by convolution in Fourier space. A dimension of extent $L$ cannot robustly measure wavelengths much larger than $L$; the fundamental wavenumber of a periodic box of that size is $2\pi /L$.

Sampling with a homogeneous Poisson point process of mean comoving number density $\bar{n}$ adds a constant $1/\bar{n}$ to the dimensional galaxy power spectrum. Subtracting its expectation leaves sampling uncertainty. Correlated tracer populations and survey weights modify this simple formula; survey depth alone does not specify $\bar{n}$.

Redshift-space distortions arise because observed galaxy redshifts include peculiar velocity as well as cosmological redshift. Distances inferred solely from expansion therefore distort clustering along the line of sight. Coherent large-scale motion and small-scale velocity dispersion produce different effects on the inferred matter power spectrum.

A galaxy red sequence is the relatively narrow red locus in a galaxy colour-magnitude diagram, populated largely by systems with little recent star formation. Its colour, abundance, and galaxy stellar mass evolve. Dust and selection must be distinguished from old stellar ages; growth can involve quenching and later assembly of existing stars.

The observed size distribution of galaxies changes with epoch at a given galaxy stellar mass. Growth of an individual object and changes in which objects enter a selected sample both contribute. Minor galaxy mergers can deposit stars at large radii, increasing size while adding relatively little mass; stellar mass loss and dissipative evolution provide other mechanisms.

Galaxy morphological evolution changes the relative prevalence and internal structure of discs, spheroids, bars, and irregular systems over time. Comparing epochs requires consistent wavelength, spatial resolution, and selection. Galaxy mergers, gas accretion and internal instabilities can all contribute.

A galaxy of characteristic radius $R$ and velocity dispersion $\sigma$ has crossing time $R/\sigma$. Gravitational balance gives $GM/R\sim {\sigma }^2$, hence $t_{\mathrm{dyn}}\sim GM/{\sigma }^3$. A numerical coefficient depends on the radius definition, density profile, and whether one uses a crossing, orbital, or free-fall time; these are not identical clocks.

Stellar asymmetric drift is the lag of a pressure-supported stellar population's mean rotation behind the local circular speed. Neglecting the tilted velocity ellipsoid, radial Jeans equation balance givesA declining density and larger random velocities generally increase the lag; gravitational balance is supplied partly by random-motion stresses.

The intracluster medium is diffuse hot plasma between the galaxies of a galaxy cluster. Its X-ray radiation includes thermal bremsstrahlung and atomic line emission. Measured temperature and density profiles supply a cluster hydrostatic mass estimator when approximately thermal hydrostatic equilibrium holds. The medium supplies much of a cluster's baryonic mass, but this need not equal its total gravitating mass.

The virial theorem balances a relaxed galaxy cluster's orbital kinetic energy, of order $M{\sigma }_v^2$, against its Newtonian gravitational potential energy, of order $-G{M}^2/R$. Their ratio gives the displayed mass scale. The numerical coefficient depends on the density profile, radius definition and conversion from line-of-sight velocity dispersion to three-dimensional orbital motion. Membership contamination, anisotropy and nonequilibrium dynamics can bias the inference.

Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

A plane-symmetric, self-gravitating stellar population with constant vertical velocity dispersion satisfies the vertical Jeans equation and the Poisson equation for Newtonian gravity. With ${\Phi }^{\prime }(0)=0$, their solution isIts surface density is $\Sigma =2{\rho }_0{z}_0$, so $z_0={\sigma }_z^2/(\pi G\Sigma )$. Its outer density tail is exponential with scale height $z_0/2$; its central profile is not exponential.