For smooth $h$ on $[0,2\pi ]$ and positive integer $n$, repeatedly integrating $\int h(t)\cos (nt)dt$ twice eliminates the sine endpoint terms and yields only even inverse powers of $n$. The exact remainder after $M$ pairs is $(-1{)}^M{n}^{-2M}\int h^{(2M)}\cos (nt)$, which is $o(n^{-2M})$ by the Riemann-Lebesgue lemma. Without integer frequencies an $n^{-1}$ sine endpoint term generally remains.

Perturbation theory constructs approximations to a parameter-dependent problem by expanding about a simpler limiting problem. In a regular perturbation, substituting $u=u_0+ϵu_1+\cdots$ gives successive equations for corrections. A singular perturbation can change the relevant scale, lose an endpoint condition or send a solution branch to infinity; then a single power series in the original variables may miss essential behavior.^[ref]

For a concrete regular perturbation, solve $u+ϵu^2=1$ near $u=1$. Matching powers gives $u_0=1$, $u_1=-1$ and $u_2=2$, soThe exact nearby root $2/(1+\sqrt{1+4ϵ})$ verifies these coefficients. The other root behaves as $-1/ϵ$ and cannot be captured by this same finite-root expansion. Thus a perturbation calculation should specify the solution branch, parameter limit and error scale; a formal hierarchy does not by itself prove convergence or uniform accuracy.

Suppose a continuous function $F(\lambda )$ on $[0,\infty )$ has a known power-law tail $F(\lambda )\sim a{\lambda }^{-p}$. Put $u=\lambda /(1+\lambda )$ and $h(u)=(1+\lambda {)}^{p}F(\lambda )$. When $h$ extends continuously to $u=1$, polynomial approximation on $[0,1]$ gives a global approximation $F(\lambda )\approx (1+\lambda {)}^{-p}{h}_N(u)$ using only elementary operations. Taylor series at zero and the tail asymptotic expansion can constrain endpoint coefficients. Agreement at the two endpoints alone does not guarantee accuracy in the intervening interval; uniform error control or intermediate calibration is needed.

An asymptotic power series uses powers as its asymptotic sequence. Near a finite point $a$, the condition in the title applies along the specified domain for every fixed $N$; at infinity, inverse powers are convenient. It need not be a convergent power series. Its coefficients record the successive limiting responses, as the integral example below demonstrates.

The finite-order Taylor theorem supplies an asymptotic power series for a smooth function near a point, but smoothness does not imply that its Taylor series converges to the function. Define $g(x)=e^{-1/x}$ for $x>0$ and $g(x)=0$ for $x\le 0$. Every right-hand derivative is an exponential function times a polynomial in $1/x$, tending to zero as $x\downarrow 0$. Consequently $g$ is a smooth function with every derivative zero at the origin. Its all-zero Taylor series misses $g(x)>0$ to the right. Zero and $g$ also have the same all-zero asymptotic power series on that side, despite being different functions. An asymptotic expansion can determine all its own coefficients without determining the represented function uniquely.

When $c(0)>0$ and $\lambda -1=O(ϵ)$, the change $x=ϵX$ resolves a truncated Cauchy distribution. Its leading contribution is $[ϵ\lambda c(0){]}^{-1}[\pi /2+\arctan ((\lambda -1)/(ϵc(0)))]$. It matches the nonresonant endpoint integral on one side and the full interior peak on the other. An interior peak at $x_{\ast }$ has width $ϵc(x_{\ast })/\lambda$; proximity to an endpoint must be measured against this width, not against $ϵ$ alone.

An approximation is uniform on a specified domain when its error bound holds throughout that domain as the small parameter tends to zero. In a matched asymptotic expansion, adding the inner and outer approximations and subtracting their common overlap produces a composite approximation. Checking only a fixed point away from a boundary layer does not establish uniformity.

A uniform-mass density self-gravitating cylinder has ${\Phi }_R=2\pi G{\rho }_{0}R$. Radial balance with the displayed rotation law gives $p_R={\rho }_0(k^2{R}^3-2\pi G{\rho }_{0}R)$. At central pressure ${\pi }^2{G}^2{\rho }_0^3/k^2$, pressure becomes ${\rho }_0{k}^2(R^2-R_0^2{)}^2/4$ and the free surface has zero effective gravity. The positive radius uses $|k|$ if either rotation direction is allowed.

Gravitational instability is the growth of a perturbation because its increased self-gravity overcomes the opposing support or restoring response. For a uniform nonexpanding gas, linearization gives the Jeans instability dispersion relation ${\omega }^2=c_s^2{k}^2-4\pi G\rho$: sufficiently long wavelengths grow rather than oscillate. In an expanding universe the background and Hubble friction modify the growth rate; a pressureless growing density mode can eventually become nonlinear. Rotation, magnetic stress, pressure and geometry can all alter the stability threshold in other systems.

A disk warp is a radial change of the orbital plane of successive disk annuli. In the small-tilt approximation, a complex tilt $W$ records its magnitude and azimuth. Pressure and internal torques communicate the tilt through bending waves of a Keplerian disk in the inviscid thin-disk regime. A flat tilted disk has constant $W$ even though its absolute orientation differs from the reference equatorial plane.

A Kuzmin disk is an infinitesimally thin axisymmetric astrophysical disk whose relative potential is $\psi =Gm/\sqrt{R^2+(|z|+b{)}^2}$, with $m,b>0$. Away from the disk this is a point-mass vacuum field. The jump of its vertical derivative is $-2Gmb/(R^2+b^2{)}^{3/2}$, so the Poisson equation for Newtonian gravity, written ${\nabla }^2\psi =-4\pi G\rho$, gives the displayed surface density. Integrating $2\pi R\Sigma$ gives total mass $m$.

In a thin constant-density disk with free surfaces $z=\pm H$, neglect slow radial gradients and let perturbations vary as $e^{ikr-i\omega t}$. With $h=p^{\prime }/\rho$, the local inviscid equations are $-i\omega u-2\Omega v=-ikh$, $-i\omega v+{\kappa }^{2}u/(2\Omega )=0$, $-i\omega w=-h_z$ and $iku+w_z=0$. Eliminating velocity gives $h_{zz}=k^2{\omega }^{2}h/({\omega }^2-{\kappa }^2)$. This local WKB approximation assumes $H/r\ll 1$ and $|kr|\gg 1$ and has both surface and inertial wave branches.

// Target: astrophysics.bigb

The black-hole sphere of influence is a characteristic radius where the central black hole appreciably changes surrounding stellar dynamics. Equating its Newtonian gravitational potential scale $G{M}_{\bullet }/r$ to the ambient squared velocity dispersion gives $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$. Dynamical mass measurements depend on angular resolution $r_{\inf }/D$, tracer brightness, and degeneracies in the orbital distribution, not solely on black hole mass.

For two positive measurements $F_1,F_2$ in the same photometric passband, using the same magnitude zero point, units, and weighting convention, their apparent magnitudes obeyThe logarithm takes a dimensionless ratio. An energy-weighted measurement must be compared with another energy-weighted measurement, and a photon-counting measurement with another consistently calibrated photon-counting measurement. Changing photometric passband or magnitude zero point requires the corresponding calibration, as in a colour index. ^[ref]. ^[ref].

For example, $F_2=100F_1$ gives $\Delta m=-5$: source 2 is brighter. Conversely, $\Delta m=+5$ gives $F_2/F_1=0.01$, and $\Delta m=+1$ gives $F_2/F_1\simeq 0.398107$. Even fractional steps work: $\Delta m=+0.1$ gives $F_2/F_1\simeq 0.912011$, an approximately $8.80\%$ decrease. These are illustrative ratios, independent of any particular star catalogue.

An observed genetic marker tags an unobserved causal genetic variant when linkage disequilibrium correlates their allele indicators. Their squared allelic correlation $r^2$ measures prediction strength. Under standard local-alternative score-test approximations, using the marker rather than the causal allele reduces the noncentrality parameter by roughly $r^2$, suggesting a sample-size multiplier about $1/r^2$. This is an approximation requiring the same disease model and sampling design, not a universal power identity.

Complete asymptotically flat spin initial data obeying the Einstein constraints and the dominant energy condition have future-causal ADM four-momentum, under the appropriate decay and boundary hypotheses. A spinorial proof solves the Witten spinor equation and integrates the Nester two-form. Its divergence is a sum of a nonnegative spinor-gradient term and a matter-energy contraction with a future-causal Dirac current. Positivity for every asymptotic constant spinor yields the full inequality, rather than merely nonnegative energy in one frame.

For an asymptotically flat spatial metric $h$ and $k_{ij}=-g({\nabla }_{i}n,e_j)$, the displayed surface integral gives ADM spatial momentum. Together with ADM energy it forms the asymptotic energy-momentum four-vector. Reversing the sign convention for extrinsic curvature reverses the corresponding surface-integral sign.

For an asymptotically flat stationary solution in its asymptotic rest frame, the ADM mass is the gravitational mass measured by the Arnowitt-Deser-Misner energy. It is fixed by the asymptotic metric falloff, not by a coordinate parameter's name. If spatial ADM momentum is present, distinguish the asymptotic energy component from the invariant mass obtained from the energy-momentum norm. For a static Schwarzschild-Tangherlini metric in $D$ dimensions with $f=1-\mu /r^{D-3}$, the mass is $(D-2){\Omega }_{D-2}\mu /(16\pi G_D)$; the dimensional normalization is also reproduced by the Komar mass normalization in higher dimensions.

Colour each interval $[2^j,2^{j+1})\cap \mathbb{N}$ by the parity of $j$. Every infinite arithmetic progression of positive common difference d meets every sufficiently large such interval, since the interval has length greater than d. Both colours occur infinitely often. Thus neither colour class contains an infinite arithmetic progression, despite the finite-length conclusion of the Van der Waerden theorem.

For nonnegative real numbers $a_i$ and positive weights $w_i$ with ${\sum }_i{w}_i=1$, the weighted geometric mean does not exceed the weighted arithmetic mean:For positive $a_i$, this follows from Jensen's inequality and the concavity of the natural logarithm; a limit extends it to zero inputs. For two terms it gives $a^s{b}^{1-s}\le sa+(1-s)b$ when $0<s<1$.

The pointwise product of arithmetic functions multiplies two arithmetic functions at the same input: $(f\cdot g)(n)=f(n)g(n)$. Thus $(1\cdot 1)(12)=1$ and $(\mathrm{id}\cdot 1)(12)=12$, using the constant-one arithmetic function and the identity arithmetic function. These differ from the corresponding Dirichlet convolution values $6$ and $28$, because Dirichlet convolution sums over every positive integer divisor.