In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function

.

The function e^-1/x². The Taylor series is 0, but the function is not.

The Taylor series for this function is exactly zero to all orders in perturbation theory, but the function is non-zero if x ≠ 0.

The implication of this for physics is that there are some phenomena which are impossible to understand by perturbation theory, however many orders of perturbation theory we use. Instantons are an example.

See also

Taylor series