Mathematical methods in physics

Elementary methods

Partial fraction decomposition – technical skill
Integration by parts and change of variables – technical skill
Series expansion of nonanalytic and implicitly defined functions – technical skill
Hyperspherical coordinates – exactly solvable problems
Spherical trigonometry – astrometry
Discrete and integral transforms – solving difference and differential equations
Change of variables in differential expressions – changing thermodynamic variables
Finding exact differential – obtaining thermodynamic potentials

Matrix inversion, solution of equations – technical skill
Generalized eigenvalue problem – variational methods in quantum mechanics
Jordan normal form (algebraic canonic form of a matrix) – solving linear ODE
Singular value decomposition (numeric canonic form of a matrix) – population analysis in quantum mechanics
Linear least squares – fitting

Fresnel integrals – light diffraction
Jacobi elliptic functions – pendulum motion
Polygamma function – series of rational functions, e.g. order statistics of stars by visual magnitude
Polylogarithm – thermodynamic quantities of quantum ideal gases with power law spectra
Riemann zeta function and its generalizations – power sums
Bessel functions – Green's function and eigenfunctions of the Helmholtz equation
Spherical harmonics – eigenfunctions of the Laplace operator on a sphere, multipole expansions
Generalized Laguerre polynomials – eigenfunctions of the quantum Kepler problem
Hermite polynomials – eigenfunctions of the harmonic oscillator
Hypergeometric functions – solutions of second order linear equations, factorial and power series

Asymptotic expansion – asymptotics of integrals and solutions of differential equations, perturbation theory in quantum mechanics, low-temperature series in statistical physics
Euler-Maclaurin formula – high-temperature series of the statistical sum
Method of steepest descent – asymptotics of integrals

First-order quasilinear equation – Hamilton-Jacobi equation, motion of a quantum particle in a homogeneous field
Parabolic equation – transport equation (diffusion, heat, radiation etc)
Hyperbolic equation – waves in a continuum medium, Maxwell's equations, Schrodinger equation, Einstein field equations
Elliptic equation – stationary solutions of the above problems
Spectral problem – time-independent Schrodinger equation