Advanced Mathematical Physics – Key Vocabulary

Essential vocabulary drawn from lecture notes on complex analysis, integral equations, calculus of variations, Green’s functions, and group theory.

Complex Variable (z)

An ordered pair of real numbers (x, y) written as z = x + iy.

Analytic (Holomorphic) Function

A complex function that is differentiable at every point in some neighbourhood; equivalently, it satisfies the Cauchy-Riemann equations with continuous partial derivatives.

Entire Function

A function that is analytic at every point of the complex plane (e.g., any polynomial, e^z).

Cauchy–Riemann Equations

For f(z)=u(x,y)+iv(x,y): ∂u/∂x = ∂v/∂y and ∂u/∂y = –∂v/∂x. They are the necessary and sufficient conditions for analyticity (with continuous partials).

Cauchy’s Integral Theorem

If f is analytic in a simply-connected region and C is any closed contour in it, then ∮_C f(z) dz = 0.

Cauchy’s Integral Formula

For analytic f inside and on a closed contour C and any point z₀ inside C: f(z₀)= (1/2πi)∮_C f(z)/(z–z₀) dz.

Laurent Series

Expansion of a function analytic in an annulus: f(z)=∑{n=0}^∞ an (z−z₀)^n + ∑{n=1}^∞ bn (z−z₀)^{−n}.

Isolated Singularity

A point z₀ where a function fails to be analytic but is analytic in some deleted neighbourhood around z₀.

Removable Singularity

An isolated singularity whose Laurent series has no principal part (all b_n = 0); the function can be made analytic by redefining f(z₀).

Pole (Order m)

An isolated singularity where only a finite number of negative-power terms exist; if bm ≠ 0 and b{m+1}=0 it is a pole of order m.

Essential Singularity

An isolated singularity with infinitely many negative-power terms in the Laurent series; the function exhibits dense behaviour near the point (Casorati–Weierstrass).

Residue

The coefficient b₁ of (z−z₀)^{−1} in the Laurent series of f around z₀; written Res(f;z₀).

Cauchy’s Residue Theorem

For f analytic except for isolated poles inside closed contour C: ∮_C f(z) dz = 2πi ∑ Residues of f inside C.

Jordan’s Lemma

A contour-integration result that ensures the vanishing of integrals over large semicircles for functions multiplied by e^{ikz} (k>0).

Neumann Series

Series solution of a Fredholm equation of 2nd kind: φ = f + λKf + λ²K²f + … ; converges if |λ|<1/|λ_min|.

Kernel (Integral Equation)

The known function K(x,t) appearing under the integral sign in an integral equation.

Fredholm Integral Equation (2nd Kind)

φ(x)=f(x)+λ∫_a^b K(x,t)φ(t) dt with fixed limits a,b.

Volterra Integral Equation (2nd Kind)

φ(x)=f(x)+λ∫_a^x K(x,t)φ(t) dt; upper limit is the variable x.

Homogeneous Integral Equation

Integral equation with f(x)=0; non-trivial solutions exist only for discrete eigenvalues λ.

Degenerate (Separable) Kernel

Kernel expressible as finite sum K(x,t)=∑{j=1}^n Mj(x)N_j(t), allowing reduction to linear algebraic equations.

Green’s Function (1-D)

Function G(x,t) satisfying LG(x,t)=−δ(x−t) with the same boundary conditions as the differential operator L; used to solve Ly=f.

Dirac Delta Function δ(x−t)

Generalised function with ∫_{−∞}^{∞}δ(x−t)g(x) dx = g(t); zero everywhere except x=t, integral equals 1.

Wronskian

For two functions u,v: W=uv'−u'v; used to build Green’s function and test linear independence.

Abel’s Identity

For second-order linear ODE p(x)y''+q(x)y'+r(x)y=0, the Wronskian satisfies W'=−(q/p)W.

Sturm–Liouville Operator

L[y]=(d/dx)[p(x)dy/dx]+q(x)y with weight w(x); yields orthogonal eigenfunctions.

Rayleigh–Ritz Method

Variational technique approximating lowest eigenvalues by minimising functional F[ψ]=∫(pψ'^2−qψ^2)dx/∫wψ^2dx over trial functions.

Functional (Calculus of Variations)

A mapping J[y]=∫_{x1}^{x2} f(y,y',x) dx whose stationary value yields differential equations.

Euler–Lagrange Equation

The condition δJ=0 leads to d/dx(∂f/∂y')−∂f/∂y = 0.

Lagrange Multiplier (Variational)

A parameter introduced to extremise a functional subject to constraints: δ(J+λ constraint)=0.

Hamilton’s Principle

Real motion of a system makes the action S=∫_{t1}^{t2}L(q,ẋ,t) dt stationary; yields Lagrange’s equations.

SU(2) Group

Group of 2×2 unitary matrices with determinant 1; has three real parameters and generators σ_i/2 (Pauli matrices).

Pauli Matrices σ_i

σ₁=[[0,1],[1,0]], σ₂=[[0,−i],[i,0]], σ₃=[[1,0],[0,−1]]; satisfy [σi,σj]=2iε{ijk}σk.

SO(3) Group

Group of 3-D real orthogonal matrices with determinant 1; describes ordinary rotations in R³.

SU(2)–SO(3) Homomorphism

Two-to-one mapping: each rotation in SO(3) corresponds to two opposite elements ±U in SU(2).

Isospin (SU(2) Flavor)

Treats proton and neutron as two states of an SU(2) doublet with I=½, I₃=±½; strong interactions conserve isospin.

SU(3) Flavor (Eight-Fold Way)

Approximate symmetry classifying hadrons into multiplets (octet, decuplet) using quark flavors u,d,s; generators are the eight Gell-Mann matrices λ_i.

Abelian Group

Group in which the operation is commutative: g₁g₂=g₂g₁ for all elements.

Homomorphism

A structure-preserving map φ:G→H with φ(g₁g₂)=φ(g₁)φ(g₂); becomes an isomorphism if bijective.

Cyclic Group

Group generated by a single element; every element is some power of the generator.

Coset

For subgroup H of G and element g, the set gH={gh | h∈H}; left cosets partition the group.

Normal Subgroup

Subgroup N such that gN=Ng for all g∈G; necessary for quotient (factor) groups.

Factor (Quotient) Group

The set of cosets G/N with operation (g₁N)(g₂N)=g₁g₂N; defined when N is normal.

Laplace Transform

Integral transform F(s)=∫₀^∞ e^{−st}f(t) dt; converts differential equations to algebraic ones.

Fourier Transform

F(ω)=∫_{−∞}^{∞}f(t)e^{−iωt}dt; represents functions as superpositions of plane waves.

Hankel Transform

For radial problems: Hn(k)=∫₀^∞ f(r)Jn(kr)r dr where J_n is a Bessel function.

Mellin Transform

M(s)=∫₀^∞ t^{s−1}f(t) dt; useful for scale-invariant problems.

Generating Function (Orthogonal Polynomials)

A series ∑ an Pn(x) that encodes all polynomials Pn; e.g., (1−2xt+t²)^{−½}=∑{n=0}^ P_n(x)t^n for Legendre polynomials.

Jordan’s Lemma (Contour Integration)

Ensures that integrals of e^{ikz}f(z) over large semicircles vanish when f is bounded and k>0.

Residue at Infinity

Defined as −Res(f;∞)=−(coefficient of 1/z in Laurent expansion at ∞); sum of all residues including ∞ is zero.

Quadratic Form of Green’s Function

For self-adjoint L, G(x,t)=∑ φn(x)φn(t)/λn where {φn} are orthonormal eigenfunctions and λ_n ≠0.

Wronskian Discontinuity Condition

Across x=t, derivative of 1-D Green’s function jumps by 1/p(t): ∂G/∂x|{t+}−∂G/∂x|{t−}=1/p(t).

Euler–Lagrange for Several Variables

For u(x,y,z): ∂f/∂u − ∂/∂x (∂f/∂ux) − ∂/∂y (∂f/∂uy) − ∂/∂z (∂f/∂u_z)=0.

Sturm–Liouville Boundary Conditions

At endpoints a,b: (α₁y+α₂y')|{a}=0 and (β₁y+β₂y')|{b}=0, ensuring self-adjointness.

Hamiltonian Operator (H)

In quantum mechanics, the energy operator; eigenvalue equation Hψ=Eψ leads to time-independent Schrödinger equation.

SU(3) Gell-Mann Matrices

Eight 3×3 traceless Hermitian matrices λ₁…λ₈ generating su(3); generalise Pauli matrices.

Eight-Fold Way

Gell-Mann–Ne’eman classification of baryons and mesons into SU(3) multiplets (octets, decuplets).

Liebnitz Rule for Variable Limits

d/dx ∫{a(x)}^{b(x)} f(x,t)dt = ∫{a}^{b} ∂f/∂x dt + f(x,b)b'(x) − f(x,a)a'(x).

Binomial Series

(1+x)^α = ∑_{n=0}^{∞} (α choose n) x^n, valid for |x|<1.

Jordan Canonical Form

Matrix representation showing eigenvalues along diagonal and possibly ones on super-diagonal; useful in studying linear operators.

Simple Pole

Pole of order 1; residue equals lim_{z→z₀}(z−z₀)f(z).

Contiguous Laurent Annulus

Region r₁<|z−z₀|<r₂ where Laurent expansion converges.

Kronecker Delta δ_{ij}

Equals 1 if i=j, 0 otherwise; appears in orthogonality relations.

Legendre Polynomial P_n(x)

Solution to (1−x²)Pn'' – 2xPn' + n(n+1)P_n = 0; orthogonal on [−1,1].

Bessel Function J_n(x)

Solution to x²y''+xy'+(x²−n²)y=0; arises in cylindrical problems.

Hypercharge (Y)

SU(3) quantum number: Y = 2⟨Q⟩ where Q is charge averaged over multiplet; together with I₃ labels hadrons.

Residue Calculation (Higher Order Pole)

For pole of order m at z₀: Res(f;z₀)=1/(m−1)! lim_{z→z₀} d^{m−1}/dz^{m−1}[(z−z₀)^m f(z)].

Unitary Matrix

Complex matrix U with U†U=I; preserves inner products (lengths) in complex space.

Orthogonal Matrix

Real matrix O with O^T O=I; determinant ±1, represents rotations/reflections.

Self-Adjoint Operator

Linear operator equal to its own adjoint; has real eigenvalues and orthogonal eigenfunctions.

Cyclic Integral (Contour Integral)

Integral over a closed path; often evaluated via residue theorem in complex analysis.

Jordan Curve

Simple closed contour in the complex plane with no self-intersections.

Simply-Connected Region

Domain in the plane with no holes; every closed contour can be continuously shrunk to a point.

Multiply-Connected Region

Region containing holes; not simply connected (e.g., annulus).

Analytic Continuation

Extension of an analytic function beyond its original domain while preserving analyticity.

Liouville’s Theorem (Complex)

Every bounded entire function is constant.

Maximum Modulus Principle

Non-constant analytic function attains its maximum modulus only on the boundary of a domain.

Laplace’s Equation

∇²φ = 0; fundamental equation for potential theory.

Poisson’s Equation

∇²φ = −ρ/ε₀ (or −f); solved using Green’s functions.