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For a harmonic function on the planar annulus $a<r<b$, the radial amplitude of a sine or cosine Fourier mode of positive integer index $n$ is a linear combination of $r^n$ and $r^{-n}$. If its values at $a,b$ are $A_n,B_n$, respectively, the displayed hyperbolic-sine formula gives that combination. It follows either by solving the two boundary equations or by putting $s=\log (r/a)$, which changes the radial equation to $R_{ss}-n^{2}R=0$. The zero angular mode is instead affine in $\log r$:Superposing the modes gives the harmonic function with suitable prescribed Dirichlet boundary conditions. For a fixed interior radius, high angular modes from either boundary are exponentially suppressed. Piecewise smooth boundary data are attained at continuity points, while jumps have the usual mean-value Fourier limit.

For $\mathbf{\alpha }=\mathrm{diag}({\alpha }_0,{\alpha }_0,{\alpha }_1)$, set $\gamma ={\alpha }_0/\eta$, $\delta ={\alpha }_1/{\alpha }_0$, and $q^2=k^2+l^2$. A steady nonzero Fourier mode obeys the displayed condition when the denominator is nonzero. At fixed $q>0$ and $0<\delta <1$, minimizing over $m^2\ge 0$ gives $m_{\ast }^2=(1-2\delta )q^2$ and ${\gamma }_{\min }^2=4(1-\delta )q^2$ for $\delta <1/2$; for $\delta \ge 1/2$ it gives $m_{\ast }^2=0$ and ${\gamma }_{\min }^2=q^2/\delta$. This is a boundary-constrained threshold of an anisotropic alpha-squared dynamo.

For an ideal incompressible flow rotating about the axis of the field $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, use cylindrical Fourier modes $e^{\sigma t+im\varphi }$. The azimuthal derivative of a cylindrical vector Fourier mode reduces induction to $\sigma \mathbf{b}=imJ\mathbf{u}/2$. If the boundary eigenproblem gives $\lambda =i\delta \nu$, with real $|\delta |<1$, eliminating the perturbation gives the displayed dispersion relation. Here $J=\nabla \times \mathbf{B}\cdot \hat{\mathbf{z}}={\mu }_0{J}_{\mathrm{phys}}$ for SI current density; the formula is conditional on that boundary eigenrelation.

In the flat slicing of de Sitter spacetime, $a\propto -1/\eta$, $\eta <0$, a Fourier mode of a minimally coupled massless scalar field obeys $f_k^{\prime \prime }-2f_k^{\prime }/\eta +c^2{k}^2{f}_k=0$. Substitution verifies the displayed positive-frequency branch. The Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$, a different variable satisfying $u_k^{\prime \prime }+(c^2{k}^2-2/{\eta }^2)u_k=0$. For nonzero $k$, the opposite-frequency branch completes the time solution; at $k=0$ the solutions are $1$ and ${\eta }^3$. This separation avoids losing the homogeneous decaying mode by taking a restricted ansatz literally at zero wave number.

For a real $C^1$ phase with continuous monotone derivative and $|f^{\prime }|\le \delta <1$, the sum-integral discrepancy has the displayed uniform bound. Periodization and the Dirichlet-Jordan convergence theorem express it as symmetric nonzero Fourier modes. Integration by parts gives endpoint terms and reciprocal-derivative variations. Monotonicity bounds their total variation by ${\sum }_{h\ne 0}2\delta /(h^2-{\delta }^2)=O((1-\delta {)}^{-1})$. Pairing the leading $1/h$ endpoint terms reduces them to the uniformly bounded sine series. The exclusion of integer nonzero frequencies is essential.

A vector cosmological perturbation is the transverse-vector sector of linear metric and matter perturbations. For a Fourier mode with nonzero wavevector $\mathbf{k}$, transversality removes the component parallel to $\mathbf{k}$, leaving two helicity components. In a gauge with $d{s}^2=a^2[-d{\eta }^2+2B_{i}d\eta d{x}^i+{\delta }_{ij}d{x}^{i}d{x}^j]$, ${\partial }_i{B}_i=0$ describes the vector metric perturbation.

Within an instantaneous vertically averaged transport closure, the local Keplerian viscous diffusion equation has effective diffusion coefficient $3{\partial }_{\Sigma }(\bar{\nu }\Sigma )$. A positive coefficient smooths short radial Fourier modes in the allowed thin-disc wavelength range; a negative coefficient gives the negative-diffusion criterion for viscous disk instability. A zero coefficient is linearly degenerate and does not establish strict decay. This local response criterion alone does not settle thermal, self-gravitating or boundary-driven stability.

A wave packet is a localized superposition of Fourier modes. In a narrow packet, nearby wavenumbers interfere to produce a slowly varying wave envelope around a rapid carrier wave. Its leading translation is governed by group velocity; the carrier's wave phase follows phase velocity. Localized does not necessarily mean compact support: a Gaussian function decays rapidly but extends over the whole line. The Gaussian envelope in a Klein-Gordon medium gives an original example with both translation and spreading.

The spatially constant Fourier mode. On a periodic domain its coefficient represents the spatial mean, up to the chosen Fourier transform normalization. Derivatives annihilate this mode, so homogeneous periodic diffusion preserves it.

In an incompressible shearing sheet with background velocity $-2Ax{\mathbf{e}}_y$, take both the perturbation velocity and the entire magnetic field to be real single Fourier modes with a common real wavevector and transverse amplitudes. Then every quadratic directional derivative vanishes, including terms involving complex conjugates. There is no uniform background magnetic field and consequently no linear magnetic-tension coupling between the amplitudes. Advecting the phase requires ${\dot{k}}_x=2A{k}_y$, ${\dot{k}}_y={\dot{k}}_z=0$. The velocity amplitude obeys $\dot{\tilde{\mathbf{v}}}-2A{\tilde{v}}_x{\mathbf{e}}_y+2\mathbf{\Omega }\times \tilde{\mathbf{v}}=-i\mathbf{k}\tilde{\psi }-\nu k^2\tilde{\mathbf{v}}$; the magnetic amplitude obeys the displayed equation. The modified pressure includes magnetic pressure; the physical pressure can contain a mean and second harmonic even when the modified pressure is a single mode.

For a minimally coupled massless scalar field in a flat radiation-dominated universe, conformal time gives $a=\alpha \tau$ and $(a\varphi {)}^{\prime \prime }-{\nabla }^2(a\varphi )=0$. Every nonzero Fourier mode is therefore a free oscillator divided by $a$. The zero-wavenumber equation instead gives $a{\varphi }_0=A+B\tau$, hence the displayed constant-plus-decaying solution. Substituting $k=0$ into a fixed-amplitude oscillatory formula retains only the $1/a$ term and loses the constant field solution. A truly general solution must include this zero mode or explicitly exclude a homogeneous background.