Path integral

Path integral

This is not about "path integrals" in the sense which was studied by Richard Feynman. See Functional integration.

In mathematics, a path integral is an integral where the function to be integrated is evaluated along a path or curve. Various different path integrals are in use. In the case of a closed path it is also called a contour integral.

Table of contents

1 Complex analysis
2 Vector calculus
3 Quantum mechanics

Complex analysis

The path integral is a fundamental tool in complex analysis,. Suppose U is an open subset of C, γ : [a, b] → U is a rectifiable curve and f : U → C is a function. Then the path integral

may be defined by subdividing the interval [a, b] into a = t₀ < t₁ < ... < t_n = b and considering the expression

The integral is then the limit as the distances of the subdivision points approach zero.

If γ is a continuously differentiable curve, the path integral can be evaluated as an integral of a function of a real variable:

When γ is a closed curve, that is, its initial and final points coincide, the notation

is often used for the path integral of f along γ.

Important statements about path integrals are given by the Cauchy integral theorem and Cauchy's integral formula.

Vector calculus

fill in

Quantum mechanics

fill in