- 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

- Probability theory – statistical physics
- Markov chains – kinetic equation
- Stochastic processes – nonequilibrium statistical physics
- Group theory – use of symmetry
- Python programming language