Mathematical Methods for Physics

The mathematical toolkit every physicist needs — from complex analysis and contour integration through differential equations, Green's functions, group theory, and special functions — with full derivations and worked examples throughout.

Course Overview

Mathematical methods form the backbone of theoretical physics. This course develops the essential mathematical tools — complex analysis, differential equations, Green's functions, group theory, and special functions — with an emphasis on applications to quantum mechanics, electrodynamics, and statistical physics. Every technique is motivated by physical problems and derived rigorously, following the tradition of Arfken-Weber, Boas, Byron-Fuller, and Hassani.

What You Will Learn

● Complex analysis: analytic functions, Cauchy's theorem, residues, conformal maps
● Differential equations: ODEs, PDEs, separation of variables, Green's functions
● Fourier analysis: transforms, series, Sturm-Liouville eigenvalue problems
● Group theory: finite groups, Lie groups & algebras, representations, symmetry in physics
● Special functions: Bessel, Legendre, hypergeometric, orthogonal polynomials
● Advanced topics: asymptotic methods, variational calculus, integral equations

Key Equations

Cauchy Integral Formula: $f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z - z_0}\,dz$

Green's Function: $\mathcal{L}\,G(\mathbf{r},\mathbf{r}') = \delta(\mathbf{r} - \mathbf{r}')$

Lie Algebra Commutator: $[T_a, T_b] = i f_{abc}\,T_c$

Bessel Equation: $x^2 y'' + x y' + (x^2 - n^2)y = 0$

Legendre Equation: $(1-x^2)y'' - 2xy' + \ell(\ell+1)y = 0$

Hypergeometric: $ {}_2F_1(a,b;c;z) = \sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}$

Fourier Transform: $\tilde{f}(k) = \int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx$

Euler-Lagrange: $\frac{\partial \mathcal{L}}{\partial y} - \frac{d}{dx}\frac{\partial \mathcal{L}}{\partial y'} = 0$

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Part I: Complex Analysis

Analytic functions, Cauchy's theorem, contour integration, residue theorem, conformal mapping, and applications to physics.

4 chapters →

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Part II: DEs & Green's Functions

Ordinary and partial differential equations, Green's functions, Fourier methods, and Sturm-Liouville eigenvalue theory.

4 chapters →

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Part III: Group Theory

Finite and continuous groups, Lie groups & algebras, representation theory, and symmetry applications in quantum mechanics.

4 chapters →

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Part IV: Special Topics

Special functions, asymptotic expansions, variational calculus, and integral equations with physical applications.

4 chapters →

Mathematical Physics & PDE — Research Seminars

A curated set of research talks covering nonlinear PDEs, dispersive equations, wave turbulence, gravitation, and singular limits — the modern frontier of mathematical analysis. Useful background for the chapters in Parts III–IV and as a window into the questions actively pursued by leading analysts.

Gravitation & Black Holes

Seminar · Mathematical Physics

Thibault Damour — Open Issues in Gravitation