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 ε), like the following:

In this example, A₀ would be the known solution to the exactly solvable initial problem and represent the "higher orders" which are found iteratively by some systematic procedure. For small ε these higher orders become successively less important.

Examples
Perturbation theory has its roots in 17th century celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Curiously, it was the need for more and more epicycles that eventually led to the 16th century Copernican revolution in the understanding of planetary orbits. The development of basic perturbation theory for differential equations was fairly complete by the middle of the 19th century. It was at that time that Charles-Eugène Delaunay was studying the perturbative expansion for the Earth-Moon-Sun system, and discovered the so-called "problem of small denominators". Here, the denominator appearing in the n'th term of the perturbative expansion could become arbitrarily small, causing the n'th correction to be as large or larger than the first-order correction. At the turn of the 20th century, this problem led Henri Poincare to make one of the first deductions of the existence of chaos, or what is prosaically called the "butterfly effect": that even a very small perturbation can have a very large effect on a system.
Perturbation theory saw a particularly dramatic expansion and evolution with the arrival of quantum mechanics. Although perturbation theory was used in the semi-classical theory of the Bohr atom, the calculations were monstrously complicated, and subject to somewhat ambiguous interpretation. The discovery of Heisenberg's matrix mechanics allowed a vast simplification of the application of perturbation theory. Notable examples are the Stark effect and the Zeeman effect, which have a simple enough theory to be included in standard undergraduate textbooks in quantum mechanics. Other early applications include the fine structure and the hyperfine structure in the hydrogen atom.
In modern times, perturbation theory underlies much of quantum chemistry and quantum field theory. In chemistry, perturbation theory was used to obtain the first solutions for the helium atom.
In the middle of the 20'th century, Richard Feynman realized that the perturbative expansion could be given a dramatic and beautiful graphical representation in terms of what are now called Feynman diagrams. Although originally applied only in quantum field theory, such diagrams now find increasing use in any area where perturbative expansions are studied.
A partial resolution of the small-divisor problem was given by the statement of the KAM theorem in 1954. Developed by Andrey Kolmogorov, Vladimir Arnold and Jurgen Moser, this theorem stated the conditions under which a system of partial differential equations will have only mildly chaotic behaviour under small perturbations.
In the late 20th century, broad dissatisfaction with perturbation theory in the quantum physics community, including not only the difficulty of going beyond second order in the expansion, but also questions about whether the perturbative expansion is even convergent, has led to a strong interest in the area of non-perturbative analysis, that is, the study of exactly solvable models. The prototypical model is the KdV equation, a highly non-linear equation for which the interesting solutions, the solitons, cannot be reached by perturbation theory, even if the perturbations were carried out to infinite order. Much of the theoretical work in non-perturbative analysis goes under the name of quantum groups and non-commutative geometry.

History
The standard exposition of perturbation theory is given in terms the order to which the perturbation is carried out: first order perturbation theory or second order perturbation theory, and whether the perturbed states are degenerate (that is, singular), in which case extra care must be taken, and the theory is slightly more difficult.
This section needs to be expanded to include the standard textbook examples of each of the three expansions.

Perturbation orders
This section develops, in simplified terms, the general theory for the perturbative solution to a differential equation to the first order. In order to keep the exposition simple, a crucial assumption is made: that the solutions to the unperturbed system are not degenerate, so that the perturbation series can be inverted. There are ways of dealing with the degenerate (or "singular") case; these require extra care.
Suppose one wants to solve a differential equation of the form
Dg(x) = λg(x)
where D is some specific differential operator, and λ is an eigenvalue. Many problems involving ordinary or partial differential equations can be cast in this form. It is presumed that the differential operator can be written in the form
D = D are known. That is, one has a set of solutions , labelled by some arbitrary index n, such that
.
Furthermore, one assumes that the set of solutions form an orthonormal set:

with δ_mn the Kronecker delta function.
To zeroth order, one expects that the solutions g(x) are then somehow "close" to one of the unperturbed solutions . That is,

and
.
where denotes the relative size, in big-O notation. To solve this problem, one assumes that the solution g(x) can be written as a linear combination of the :

with all of the constants except for n, where . Substituting this last expansion into the differential equation, taking the inner product of the result with , and making use of orthogonality, one obtains

This can be trivially re-written as a simple linear algebra problem of finding the eigenvalue of a matrix, where

where the matrix elements A_nm are given by

Rather than solving this full matrix equation, one notes that, of all the c_m in the linear equation, only one, namely c_n, is not small. Thus, to the first order in ε, the linear equation may be solved trivially as

since all of the other terms in the linear equation are of order . The above gives the solution of the eigenvalue to first order in perturbation theory.
The function g(x) to first order is obtained through similar reasoning. Substituting

so that

gives an equation for . It may be solved integrating with the partition of unity

to give

which gives the exact solution to the perturbed differential equation to the first order in the perturbation ε.
Several important observations can be made about the form of this solution. First, the sum over functions with differences of eigenvalues in the denominator resembles the resolvent in Fredholm theory. This is no accident; the resolvent acts essentially as a kind of Green's function or propagator, passing the perturbation along. Higher order perturbations resemble this form, with an additional sum over a resolvent appearing at each order.
The form of this solution is sufficient to illustrate the idea behind the small-divisor problem. If, for whatever reason, two eigenvalues are close so that difference become small, the corresponding term in the sum will become disproportionately large. In particular, if this happens in higher-order terms, the high order perturbation may become as large or larger in magnitude than the first-order perturbation. Such a situation calls into question the validity of doing a perturbation to begin with. This can be understood to be a fairly catastrophic situation; it is frequently encountered in chaotic dynamical systems, and requires the development of techniques other than perturbation theory to solve the problem.
Curiously, the situation is not at all bad if two or more eigenvalues are exactly equal. This case is referred to as singular or degenerate perturbation theory. The degeneracy of eigenvalues indicates that the unperturbed system has some sort of symmetry, and that the generators of the symmetry commute with the unperturbed differential equation. Typically, the perturbing term does not possess the symmetry; one says the perturbation lifts or breaks the degeneracy. In this case, the perturbation can still be performed; however, one must be careful to work in a basis for the unperturbed states so that these map one-to-one to the perturbed states, rather than being a mixture.

First-order non-singular perturbation theory
Consider the following equation for the unknown variable x:
x = 1 + εx we put:

There are 5 solutions for y₀: 0, 1, -1, i and -i. We must disregard the solution y = 0. The case y = 0 corresponds to the original regular solution which appears to be at zero for ε = 0, because in the limit we are rescaling by an infinite amount. The next term is y₁ = − 1 / 4. In terms of x the four solutions are thus given as:

Perturbation theory in chemistry

Structural stability
Eigenvalue perturbation
Cosmological perturbation theory