*This article describes perturbation theory as a general mathematical method. For perturbation theory as applied to quantum mechanics, see perturbation theory (quantum mechanics).*

**Perturbation theory** comprises mathematical
methods that are used to find an approximate solution
to a problem which cannot be solved exactly, by starting
from the exact solution of a related problem.
Perturbation theory is applicable if the problem at hand can
be formulated by adding a "small"
term to the mathematical description of the exactly solvable problem.
Perturbation theory leads to an expression for
the desired solution in terms of a power series in some "small"
parameter that quantifies the deviation from the exactly
solvable problem. The leading term in this power series is
the solution of the exactly solvable problem, while further
terms describe the deviation in the solution, due to the deviation
from the initial problem. Formally, we have for the approximation to the full solution A a series in the small parameter (here called ), like the following:

Examples for the "mathematical description" are: an algebraic equation, a differential equation (e.g. the equations of motion in celestial mechanics or a wave equation), a free energy (in statistical mechanics), a Hamiltonian operator (in quantum mechanics).

Examples for the kind of solution to be found perturbatively: the solution of the equation (e.g. the trajectory of a particle), the statistical average of some physical quantity (e.g. average magnetization), the ground state energy of a quantum mechanical problem.

Examples for the exactly solvable problems to start with: Linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).

Examples of "perturbations" to deal with: Nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/Free Energy.

For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.

Consider the following equation for the unknown variable :

For the initial problem with , the solution is . For small the lowest order approximation may be found by inserting the ansatz

The same problem occurs in many real applications in physics and elsewhere: Perturbation theory may only be used to find those solutions of a problem that evolve smoothly out of the initial solution when changing the parameter (that are "adiabatically connected" to the initial solution). In physics, this fails whenever the system may go to a different "phase" of matter, with a qualitatively different behaviour that cannot be understood by perturbation theory (e.g. a solid crystal melting into a liquid).