For the dimensionless quantum harmonic oscillator $H=-d^2/d{x}^2+x^2$, the normalized compact trial function is ${\psi }_a=\sqrt{15/(16a^5)}(a^2-x^2)$ on $|x|\le a$ and zero elsewhere. Its kinetic and potential contributions to the energy quadratic form are $5/(2a^2)$ and $a^2/7$. Minimizing their sum gives $a^4=35/2$ and the displayed upper bound, approximately $1.19523$, compared with exact ground state eigenvalue $1$. The weak derivative is square-integrable because the trial is continuous at its endpoints. Jumps in the ordinary first derivative do not invalidate the Rayleigh-Ritz variational principle: it requires the energy form domain, rather than the smaller operator domain.

On $(0,L)$ with zero Dirichlet boundary conditions, the normalized trial $\sqrt{2/L}\sin (\pi x/L)$ has kinetic Rayleigh quotient ${\pi }^2/L^2$ and mean position $L/2$ by reflection symmetry. The Rayleigh-Ritz variational principle therefore gives the displayed upper bound for a real linear potential. For $ϵ\ge 0$ the lower bound ${\lambda }_1\ge {\pi }^2/L^2$ follows from the Poincaré inequality. For $ϵ\ne 0$ the trial is not an exact eigenfunction, so the upper bound is strict.

In an 1880 paper, John William Strutt, known as Lord Rayleigh, studied the resultant of many oscillations with a common frequency and arbitrary oscillation phases. His large-number calculation distinguished a mean intensity from a definite resultant amplitude and used planar coordinates followed by a polar integral. The resulting amplitude density is the Rayleigh distribution in a different scale convention. This is a large-number description, not an exact law for every finite collection of oscillations.^[ref]

To see the connection in a chosen model, take independent oscillation phases ${\Theta }_j$ with uniform distribution on $[0,2\pi )$ and equal component amplitude $a$. The phasor sum has coordinatesBoth have mean zero, variance $n{a}^2/2$ and zero covariance. The multivariate central limit theorem gives an approximately circular Gaussian random vector for large $n$, with $\sigma =a\sqrt{n/2}$. Its amplitude $\sqrt{X_n^2+Y_n^2}$ is therefore approximately Rayleigh. For every finite $n$, that amplitude is bounded by $na$, whereas a Rayleigh distribution has an unbounded tail. At $n=1$ it is exactly $a$. This simple endpoint check prevents the approximation from becoming a false exact identity.

Using $u=r^2/(2{\sigma }^2)$ in the defining expected value integral givesThe last integral is the gamma function. Near zero, the integrand before substitution behaves as $r^{q+1}$, explaining the threshold $q>-2$; at infinity the Gaussian tail controls every power. In particular,^[ref]

For the chosen $\sigma =2\mathrm{cm}$, the mean radius is about $2.5066\mathrm{cm}$, its median is $2\sqrt{2\log 2}\simeq 2.3548\mathrm{cm}$, and its mode is $2\mathrm{cm}$. Its root mean square is $2\sqrt{2}\simeq 2.8284\mathrm{cm}$. Different summaries answer different questions; the component scale $\sigma$ is neither the radial mean nor the radial standard deviation.

Squaring a Rayleigh distribution variable produces an exponential distribution. For $w\ge 0$,where the exponential distribution parameter is a rate. Equivalently, $R^2/{\sigma }^2$ has chi-squared distribution with two degrees of freedom (statistics). An amplitude and its square therefore have different probability density functions, even though one determines the other.

This makes the Rayleigh-normal scale mixture transparent: at unit scale, $R^2$ is exponential, allowing a random normal scale to produce a Laplace distribution. Another equivalent parametrization uses a Weibull distribution with shape two and scale $\sqrt{2}\sigma$. Conventions that call this latter scale the Rayleigh scale differ by a factor of $\sqrt{2}$; check the probability density function before comparing numerical parameter values.^[ref]

For $r\ge 0$, integration gives the survival function and cumulative distribution functionSolving $F_R(r_p)=p$ gives the quantile function^[ref]

For the chosen positioning model $\sigma =2\mathrm{cm}$,The disk of radius one component standard deviation therefore contains about 39.35% of outcomes; the familiar one-dimensional normal interval percentage does not apply to a two-dimensional disk. The 95% and 99% radial thresholds areA circle defined by $R=r_p$ itself still has probability zero; its enclosed disk carries the stated probability.

The closed-form quantile function also gives inverse transform sampling: if $U$ has uniform distribution on $(0,1)$, then $\sigma \sqrt{-2\log U}$ has Rayleigh distribution. Using $1-U$ instead gives the same law. Endpoints are excluded so that the logarithm stays finite.

The joint probability density of $X,Y$ isIn polar coordinates, $x=r\cos \theta$, $y=r\sin \theta$, and the absolute Jacobian determinant is $r$. Thus the joint density of radius and angle factors asIntegrating over angle gives the Rayleigh distribution. The same factorization shows that the angle has uniform distribution and is independent of the radius. This is a short change of variables argument: an annulus of width $dr$ has area $2\pi rdr$ to first order.

The assumptions are substantive. Unequal component variances give elliptical rather than circular density contours; a nonzero mean displaces the center. Even zero covariance and equal normal distributions do not suffice without independence. For an original counterexample, let $X\sim N(0,{\sigma }^2)$ and let $S$ be an independent Rademacher random variable. Put $Y=SX$. Then $Y$ has the same normal distribution as $X$, and $\mathrm{Cov}(X,Y)=\mathbb{E}[S{X}^2]=0$. Nevertheless, all points lie on $y=x$ or $y=-x$, and $R=\sqrt{2}|X|$ has densitywhich is positive at zero, unlike the Rayleigh distribution. This pair is not a Gaussian random vector.

Let $a,b,c\in {\mathbb{R}}^n$, with $c\ne 0$. The equation $x=a+(b\cdot x)c$ is the linear system $(I-c{b}^T)x=a$, a rank-one matrix update of the identity matrix. Every solution has $x=a+\lambda c$. Substitution reduces the entire problem toIf $b\cdot c\ne 1$, there is one solution, $x=a+c(b\cdot a)/(1-b\cdot c)$. If $b\cdot c=1$ and $b\cdot a=0$, every $\lambda \in \mathbb{R}$ works and the solution set is the straight line $a+\mathbb{R}c$. If $b\cdot c=1$ and $b\cdot a\ne 0$, there is no solution.

The first reduction follows directly from the original equation in any dimension. In three dimensions it can also be obtained by taking the cross product with $c$, giving $(x-a)\times c=0$. The singular denominator therefore represents two different possibilities, inconsistency or a family of solutions, and must not simply be discarded as an undefined formula.

Suppose $A\ne 0$ and $R>0$, and $H$ is an analytic function on a disk of radius strictly larger than $R$. The Taylor series at zero of $f(z)=A\log (1-z/R)+H(z)$ has radius of convergence $R$. Its coefficients are the sum of $-A/(n{R}^n)$ and the more rapidly decreasing coefficients of $H$. At $x=-R$, the logarithmic part is an alternating harmonic series and converges conditionally; at $x=R$ it diverges. The power series for $H$ converges absolutely at both endpoints and therefore cannot change either conclusion. The real convergence interval is $[-R,R)$.

For $f(z)=\log (2-e^z)$, take $R=\log 2$ and $H(z)=\log [(2-e^z)/(1-z/R)]$, with its branch normalized by $H(0)=0$. The quotient has nonzero removable value $2R$ at $z=R$. Its next zeros are at $R\pm 2\pi i$, so a holomorphic logarithm exists on a larger disk. This shows why the Taylor series converges at $x=-\log 2$ even though its singular positive endpoint is excluded.

Write the Maclaurin series coefficients as $a_n=f^{(n)}(0)/n!$. When the successive ratios exist eventually and have a limit, the ratio test gives the displayed radius of convergence, including zero and infinity. The unconditional formula is $R^{-1}={\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}$, from the Cauchy-Hadamard theorem. Endpoints must be checked separately.

The existence and convergence of this power series do not by themselves imply that it equals $f$: equality requires analyticity or control of the Taylor remainder. The smooth function $f(x)=e^{-1/x^2}$ for $x\ne 0$, $f(0)=0$, has all derivatives zero at zero, so its Maclaurin series is identically zero despite the nonzero values away from zero.

A complex visibility is a calibrated complex correlation of the signals received at a particular interferometric baseline. In a narrow field, with scalar radiation, the far-field approximation for an outgoing source, a narrow frequency channel, negligible noncoplanar phase curvature and a steady sky of mutually incoherent emitters,Here $l,m$ are small dimensionless sky-direction offsets, $I$ is brightness in the chosen channel, and $A$ is the primary beam. This is the Fourier transform of beam-weighted brightness. The two-dimensional formula requires those approximations: a wide field generally needs the third baseline coordinate and curved sky geometry.

For an explicit one-dimensional derivation, set the primary beam to one over the small field, and place independent emitters at angular offsets ${\theta }_j$. Choose a wave phase convention in which, after removing the phase centre's delay, their received complex amplitudes areHere ${\delta }_{jk}$ is the Kronecker delta and angle brackets denote an expected value estimated by time averaging a steady signal. The last equality states mutual incoherence: different emitters have zero averaged cross term, although each emitter's two received copies remain correlated. Multiplying and averaging givesFor continuous brightness this becomes $V(u)=\int I(\theta )e^{-2\pi iu\theta }d\theta$. Real brightness implies the Hermitian symmetry $V(-u)=V(u{)}^{\ast }$; it does not require brightness to be even. In particular, an asymmetric real source usually has a nonzero imaginary complex visibility. At zero baseline, $V(0)={\sum }_j{I}_j$ is the total beam-weighted source strength. Ordinary separated-element correlations need not measure that zero-baseline value.

Radio interferometer calibration estimates instrumental and propagation effects that alter a complex visibility. A simple narrow-field model iswhere $g_1,g_2$ are complex receiver gains and $n_{12}$ represents additive noise. A point-like astronomical source of known position and stable strength constrains these gains: its expected complex visibility has a known wave phase and amplitude. A resolved or variable reference source requires a better source model. Errors in the reference model can be transferred into the inferred sky.

In a proposed extension, Martin Ryle described simultaneous observation of a nearby reference point source to correct atmospheric wave phase errors, using an additional dish at each element. He presented this as a possible extension, rather than the standard One-Mile Telescope arrangement.^[ref]

An interferometric baseline is the displacement $\mathbf{b}={\mathbf{r}}_2-{\mathbf{r}}_1$ between two antennas. For a unit line of sight ${\mathbf{s}}_0$ toward the phase centre, its projection onto the plane of the sky isHere $c$ is the speed of light. The signed delay depends on which receiver is delayed; the geometric path difference's magnitude does not. If ${\mathbf{e}}_{\theta }$ is a unit tangent in the direction being imaged, the dimensionless coordinate $u=\mathbf{b}\cdot {\mathbf{e}}_{\theta }/\lambda$ is a spatial frequency, measured in cycles per radian of small sky offset. A single pair gives fringes of angular period $1/|u|$ along that tangent. This period specifies one measured oscillation; it does not specify a whole image's angular resolution or eliminate ambiguous source arrangements.

The left panel separates the component producing the reference delay from the component measuring angular structure. The right panel shows a possible Earth-rotation aperture synthesis track, with its opposite complex conjugate track. Those opposite samples provide the symmetry required by a real brightness distribution; they are not independent extra observations.

The phase centre is the reference sky direction for removing geometric delay and its associated wave phase. An unresolved source exactly there ideally has constant complex visibility after calibration; a displaced source has a baseline-dependent wave phase.

On a shell $0<r_{-}<r<r_{+}$ with homogeneous Dirichlet boundary conditions, put $L=r_{+}-r_{-}$ and $u=r\psi$. The radial heat equation in three dimensions becomes $u_t=\kappa u_{rr}$, with zero values at both endpoints. Expanding $u(0,r)=r\psi (0,r)$ in a Fourier sine series gives the displayed solution, with $b_n=(2/L){\int }_{r_{-}}^{r_{+}}r\psi (0,r)\sin [n\pi (r-r_{-})/L]dr$. Each radial mode decays at the square of its wave number. The positive inner radius makes division by $r$ harmless throughout the shell.

For a classical monatomic homogeneous fluid with pairwise central pair potential $v(r)$, the mean interaction energy is one half the sum over neighbours of a tagged particle. Using the radial distribution function gives the displayed formula; the factor one half prevents counting each pair twice. The kinetic term follows from the equipartition theorem. Its use requires the relevant integral to exist and a thermodynamic state specified by number density and temperature. Quantum kinetic contributions, many-body forces and internal molecular modes require modifications.

For a differentiable central pair potential in a classical homogeneous three-dimensional fluid, the mechanical virial theorem gives the displayed pressure. The radial distribution function accounts for the probability of each pair separation. In the dilute limit, $g(r)=e^{-v(r)/(k_{B}T)}+O(n)$, and integration by parts yields $B_2(T)=-2\pi {\int }_0^{\infty }[e^{-v(r)/(k_{B}T)}-1]r^{2}dr$. Thus the second virial coefficient controls $p/(n{k}_{B}T)=1+B_{2}n+\cdots$. A few virial terms are a low-density approximation, not generally a converged description of a dense liquid. A discontinuous hard core needs its contact form or a limiting argument.

A constituent quark mass is an effective mass parameter in a low-energy constituent-quark model. It accounts approximately for strong-interaction dressing and some spin-independent bound-state dynamics; it is not identical to a renormalized quark mass in the Standard Model Lagrangian.

The position uncertainty of a quantum state is the standard deviation of its measured position, provided the second moment is finite. It is computed from the position operator with the Born rule. An oscillator coherent state has $\Delta x=\sqrt{\hslash /(2m\omega )}$ independently of its mean position; translating the state changes the mean without changing this variance.

If a single approximately stationary emitting region changes coherently through a causal disturbance, a disturbance travelling no faster than the speed of light takes at least $L/c$ to coordinate a scale $L$. Light-travel differences also smear simultaneous emission from different depths. Under those assumptions, observed intrinsic variability on a time scale $\Delta t$ suggests $L\lesssim c\Delta t$.

For a chosen example $\Delta t=0.010\mathrm{s}$, multiplication by $c=299,792.458\mathrm{km}{\mathrm{s}}^{-1}$ gives $L\lesssim 2998\mathrm{km}$. This is an order-of-magnitude causality bound on a coordinated emitting scale. It is not a measured stellar radius, and even the relation between $L$ and a diameter depends on geometry. A pulsar beam crossing is a geometric clock, and pulsar dispersion delay, scattering and instrumental response can broaden its observed pulse. The recurrence period, observed pulse width and intrinsic variability time must therefore be distinguished before applying the bound. Relativistically moving emitters require further arrival-time corrections.