For real $a>0$ and $\mathrm{Re}s>0$, the change of variables $u=at$ in the defining improper integral for the Gamma function givesThe improper integral converges absolutely: near zero its absolute integrand behaves like $t^{\mathrm{Re}s-1}$, and at infinity the exponential function overcomes every fixed power.^[ref]

For real $s=1,2,4$ and $a=1$, the integrands are $e^{-t}$, $t{e}^{-t}$, and $t^3{e}^{-t}$, with total areas $\Gamma (1)=1$, $\Gamma (2)=1$, and $\Gamma (4)=6$. The third integrand is taller and has six times the area of the first, although the exponential function makes each tail decay.

For real $s>0$, the kernel $t^{s-1}{e}^{-t}$ becomes the probability density function of a Gamma distribution with shape $s$ and unit rate only after division by its normalizing constant $\Gamma (s)$. In particular, the shape-four probability density function is $t^3{e}^{-t}/6$. More generally, the Gamma integral verifies that $a^s{t}^{s-1}{e}^{-at}/\Gamma (s)$ integrates to one for shape $s>0$ and rate $a>0$.

For the half-integer evaluation, the change of variables $t=u^2$ reduces the Gamma integral to the Gaussian integral:Consequently, for real $u>0$,

A flat galaxy rotation curve has approximately constant circular speed $v_c$ over a radial interval. Under spherical symmetry it implies $M(<r)=v_c^{2}r/G$ and mass density $\rho (r)=v_c^2/(4\pi G{r}^2)$.

In a spherical potential, the circular speed satisfiesIt measures the gravitational field even when the system itself is not supported by circular motion.

Galactic chemical evolution follows the production, transport, dilution, locking, and loss of chemical elements as stars form and die and gas enters or leaves a galaxy.

A galactic outflow carries gas, metals, energy, and momentum away from a galaxy under stellar or active-galactic-nucleus feedback. Whether the material escapes or later returns depends on its speed and the galactic potential.

Stellar feedback is the transfer of energy, momentum, mass, and newly formed elements from stars to surrounding matter through radiation, winds, supernovae, and cosmic rays.

For treatment history ${\bar{A}}_t$, observed history $H_t$, and final potential outcome $Y(\bar{a})$, sequential exchangeability requiresat every treatment time. Together with consistency of potential outcomes and positivity in causal inference, it identifies longitudinal effects by G-computation.

A projection map from a Cartesian product selects specified coordinates, for example $p_j:{\prod }_i{X}_i\to X_j$ with $p_j((x_i{)}_i)=x_j$.

A regular singular point is a singularity at which solutions of a linear differential equation have at worst controlled power and logarithmic behavior after a Frobenius reduction.

Suppose every derivative $x\mapsto \varphi (x+d)-\varphi (x)$ on ${\mathbb{F}}_p^n$ takes at most $C$ values. Then $\varphi$ restricts to a Freiman homomorphism on a set of density at least $p^{-1}{C}^{-5}$. The proof bounds the second-difference image by $C^5$ and chooses a random subspace avoiding its nonzero elements.

Let $X$ be the image of all additive second differencesIf a subspace $V$ satisfies $V\cap X=\{0\}$, then the restriction of $\varphi$ to the inverse image of every coset of $V$ is a Freiman homomorphism.

A map $\varphi :A\to B$ is a Freiman $s$-homomorphism whenimplies the corresponding equality between the $\varphi (a_i)$. It is a Freiman $s$-isomorphism when it is bijective and the converse implication also holds.

Every finitely generated free group is a residually finite group. For $F_2$, the ping-pong lemma embeds it into ${\mathrm{SL}}_2(\mathbb{Z})$, and reduction modulo a prime in an integral matrix group separates each nonidentity element in a finite quotient.

Suppose nontrivial groups $G_1,G_2$ act on a set and there are disjoint nonempty subsets $X_1,X_2$ such that every nonidentity element of $G_1$ maps $X_2$ into $X_1$, while every nonidentity element of $G_2$ maps $X_1$ into $X_2$. Assume at least one factor has at least three elements. Then the subgroup generated by $G_1$ and $G_2$ is their free product. The cardinality hypothesis excludes the counterexample in which $G_1=G_2=C_2$ acts by swapping $X_1$ and $X_2$. Versions allowing both factors to have order two need an additional separation hypothesis.

For every function $f:S\to G$ from a set into a group, there is a unique group homomorphism $\bar{f}:F(S)\to G$ whose restriction to $S$ is $f$.

If a $k$-uniform set family on an $n$-element ground set has at most $s$ possible intersection sizes between distinct members, then its size is at most $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$. The proof uses the linear independence of incidence polynomials on the $k$-slice.

A proportional frailty model has conditional hazard $h(t\mid U=u)=u{h}_0(t)$. Its conditional survival function is $e^{-u{H}_0(t)}$, so its population survival function is the Laplace transform of the frailty distribution evaluated at $H_0(t)$.

For an additive character $\psi$ of a non-Archimedean local field $F$,For the character induced from the standard character of ${\mathbb{Q}}_p$, this annihilator is the inverse different.

The Fermi constant is the coefficient conventionally used for the low-energy charged-current weak interaction. Tree-level exchange of a $W$ boson gives $G_F/\sqrt{2}=g^2/(8M_W^2)$.

A two-link follower-force model replaces a filament by two rigid links with torsional springs and point drags. The nonsymmetric linearized stiffness can produce a Hopf bifurcation even without inertia.