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If $f$ has a pole at $a$, then $e^f$ has an essential singularity there. The exponential series produces infinitely many negative Laurent powers from the nonzero principal part of $f$.

A meromorphic function is holomorphic except at isolated poles. Equivalently, it is locally a quotient of two holomorphic functions whose denominator is not identically zero.

For large $\lambda$, a Poisson random variable of mean $\lambda$ is approximately normal with mean and variance $\lambda$.

Independent Poisson counts conditioned on their sum have a multinomial distribution.

If $N_{\lambda }\sim \mathrm{Pois}(\lambda )$ and $\lambda \to \infty$, thenIndeed, its characteristic function is

For $\overline{X}\mid \mu \sim N(\mu ,1/n)$ and an equal mixture of $\mu =0$ and $\mu \sim N(0,{\tau }^2)$,

The mean-zero Sobolev space is the closed subspaceOn a bounded connected Lipschitz domain, the Poincare-Wirtinger inequality makes $\Vert \nabla u{\Vert }_2$ an equivalent Hilbert norm on this space.

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives

Forthe dispersion relation is $\omega =Uk+k^9/9$, so $c_p=U+k^8/9$ and $c_g=U+k^8$.

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.

A wave crest is a local maximum of the wave's displacement. In a monochromatic travelling wave it lies at a fixed phase and propagates at the phase velocity.

For refractive indices $n_{-}$ and $n_{+}$,

The liquidus is the locus in a phase diagram above which the equilibrium state is entirely liquid. In a dilute binary solution it is often approximated locally by a linear relation such as $T_L=T_0-mC$.

Permutation cycles with disjoint supports commute. The order of their product is the least common multiple of their lengths.

A three-cycle is a permutation cycle of length three. On any chosen set of three letters, the two possible three-cycles are inverses of one another.

The paw graph $H$ has $R(H)=7$. A colouring of $K_6$ with two disjoint red triangles and all cross-edges blue avoids a monochromatic paw. In $K_7$, a monochromatic triangle exists; avoiding a paw forces all its edges to the other four vertices into the opposite colour, which then forces those four internal edges into the triangle's colour and creates a paw.

Magnetic susceptibility describes how readily a magnetic field changes a material's magnetization. The field variable and normalization matter: volume magnetic susceptibility in the International System of Units is dimensionless, whereas a derivative of total magnetic moment with respect to $B$ has units of $\mathrm{J}{\mathrm{T}}^{-2}$. A slope measured against the applied magnetic field intensity also depends on the sample's demagnetizing field.

For an isotropic, reversible linear response through the origin, writeHere $\mathbf{M}$ is magnetization, or magnetic moment per volume; ${\mathbf{H}}_{\mathrm{int}}$ is the internal magnetic field intensity; ${\mathbf{B}}_{\mathrm{int}}$ is the internal magnetic flux density; and ${\mu }_0$ is the vacuum permeability. Both $M$ and $H$ have units $\mathrm{A}{\mathrm{m}}^{-1}$. The magnetic permeability is ${\mu }_0(1+{\chi }_v)$. ^[ref]

The independent-moment magnetic response below shows why one number cannot describe every field: magnetization approaches a saturation magnetization, while its differential magnetic susceptibility approaches zero. Anisotropy requires a magnetic susceptibility tensor; hysteresis requires specifying the preparation and sweep branch as well as the bias magnetic field, temperature, and timescale. The Gouy balance turns a small response into a measurable force without presuming that its readout is already the intrinsic volume magnetic susceptibility.

The winding number is the integer represented by the pattern in the first homology of the solid torus, equivalently its algebraic intersection number with a meridional disk.