Perturbationsserien

Perturbationsserien, often translated as perturbation series, is a mathematical method used to find approximate solutions to problems that cannot be solved exactly. These problems typically involve a small parameter, denoted as epsilon, which represents a "perturbation" to a simpler, exactly solvable problem. The core idea is to express the solution as a power series in this small parameter.

The method starts with the known exact solution to the unperturbed problem (when epsilon is zero). Then, the solution to the perturbed problem is assumed to be of the form of a series: X = X0 + epsilon * X1 + epsilon^2 * X2 + ..., where X0 is the unperturbed solution, and X1, X2, etc., are subsequent correction terms. These correction terms are determined by substituting this series into the original equation and collecting terms with the same power of epsilon. By setting the coefficient of each power of epsilon to zero, a series of simpler equations are obtained, which can usually be solved iteratively.

Perturbationsserien is widely applied in various fields of science and engineering. In physics, it is crucial for solving problems in quantum mechanics, celestial mechanics (e.g., calculating the orbits of planets under gravitational influences beyond the dominant one), and fluid dynamics. In mathematics, it is used in differential equations and dynamical systems. The validity of the perturbation series depends on the smallness of the parameter epsilon; if epsilon becomes too large, the series may not converge or may provide a poor approximation.