Glossary · Mathematical Methods II
Alphabetical reference for the PHY622 wiki.
Analytic (holomorphic) — Complex-differentiable in a neighbourhood. An extraordinarily strong condition: it forces infinite differentiability, a convergent Taylor series, and the Cauchy–Riemann equations. → Analytic functions
Analytic continuation — The unique extension of an analytic function beyond its original domain. Unique because two analytic functions agreeing on a set with a limit point agree everywhere. → Singularities & branch points
Argument principle — \(\frac{1}{2\pi i}\oint f'/f\,dz\) counts zeros minus poles inside the contour. → Residue theorem
Beltrami identity — First integral of the Euler–Lagrange equation available when the integrand has no explicit independent variable: \(f - y'\partial f/\partial y' =\) const. The variational analogue of energy conservation. → Brachistochrone
Brachistochrone — The curve of fastest descent under gravity; a cycloid. Bernoulli's 1696 challenge problem. → Brachistochrone
Branch cut — A curve removed from the plane to make a multivalued function single-valued. Its location is a choice; only the branch points are intrinsic. → Singularities & branch points
Branch point — A point around which a function fails to return to its value; not an isolated singularity, and has no residue. → Singularities & branch points
Cauchy–Riemann equations — \(u_x = v_y\), \(u_y = -v_x\) for \(f = u+iv\). Necessary (and with continuity, sufficient) for analyticity; they force \(u\) and \(v\) to be harmonic. → Cauchy–Riemann
Cauchy's integral formula — \(f(z_0) = \frac{1}{2\pi i}\oint\frac{f(z)}{z-z_0}dz\): boundary values determine the interior completely. → Cauchy's integral formula
Cauchy's theorem — \(\oint f\,dz = 0\) for \(f\) analytic inside and on the contour. → Cauchy's theorem
Conformal map — An analytic map with \(f'\neq0\); preserves angles. Fails exactly where \(f' = 0\). → Conformal mapping
Conjugacy class — \(\{gxg^{-1}\}\) for fixed \(x\). Elements in the same class have the same order and, in a symmetry group, represent "the same kind of operation". → Symmetry groups
Coset — \(gH\) for a subgroup \(H\). Cosets partition the group into equal-sized blocks, which is the whole proof of Lagrange's theorem. → Cosets & Lagrange
Essential singularity — Infinitely many negative powers in the Laurent expansion. By Picard's theorem the function takes almost every complex value infinitely often nearby. → Laurent series
Euler–Lagrange equation — \(\frac{d}{dx}\frac{\partial f}{\partial y'} - \frac{\partial f}{\partial y} = 0\); the condition for a functional to be stationary. → Euler–Lagrange
Fermat's principle — Light takes a path of stationary optical length. Yields Snell's law, and is the optical twin of Hamilton's principle. → Fermat's principle
Functional — A map from functions to numbers, e.g. \(T[y] = \int L(y,y',x)\,dx\). The object the calculus of variations extremises. → Calculus of variations
Geodesic — A stationary-length curve on a surface or in a spacetime. → Geodesics
Group — A set with an associative operation, an identity and inverses. → Symmetry groups
Hamilton's principle — Physical trajectories make the action \(\int L\,dt\) stationary. All of mechanics, from one sentence. → Hamilton's principle
Homomorphism — A structure-preserving map between groups. Its kernel is always a normal subgroup. → Cosets & Lagrange
Jordan's lemma — The semicircular arc of \(\oint e^{iaz}f(z)\,dz\) vanishes for \(a>0\) if \(f\to0\). Why you integrate \(e^{iz}\) and not \(\cos z\). → Residue theorem
Lagrange's theorem — The order of a subgroup divides the order of the group. Pure counting, and strikingly restrictive. → Cosets & Lagrange
Laurent series — Expansion in an annulus allowing negative powers. Not unique — it depends on which annulus. → Laurent series
Lie group — A group that is also a smooth manifold; continuous symmetry. → Lie groups
Minimal surface — A surface of stationary area; soap films. → Minimal surfaces
Möbius transformation — \(w = (az+b)/(cz+d)\); maps generalised circles to generalised circles. → Conformal mapping
Normal subgroup — \(gH = Hg\) for all \(g\). Exactly the subgroups you may quotient by, and exactly the kernels of homomorphisms. → Cosets & Lagrange
Pole of order \(m\) — Laurent principal part terminating at \((z-z_0)^{-m}\). → Singularities & branch points
Principal value — The symmetric limit that assigns a finite value to an integral through a pole. Paired with \(\pm i\pi\,\mathrm{Res}\) in Sokhotski–Plemelj. → Residue theorem
Rayleigh–Ritz — Estimating the lowest eigenvalue by minimising the Rayleigh quotient over a trial family; always an upper bound. → Calculus of variations
Representation — A homomorphism from a group into matrices; how abstract symmetry acts on states. → Group representations
Residue — The coefficient \(a_{-1}\) of the Laurent expansion; equivalently \(\frac{1}{2\pi i}\oint f\,dz\) about the pole. → Residue theorem
Sokhotski–Plemelj — \(\frac{1}{x\mp i\epsilon} = \mathrm{P}\frac1x \pm i\pi\delta(x)\). The formula behind Landau damping. → Residue theorem
\(SU(2)\to SO(3)\) — A two-to-one homomorphism with kernel \(\{\pm I\}\); why a spinor needs 720° to return. → SU(2) → SO(3)
Tautochrone — A curve on which descent time to the bottom is independent of starting height — also the cycloid. → Brachistochrone
Transversality condition — The extra boundary condition required when an endpoint is free rather than fixed. → Calculus of variations