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.
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 = t_{0} < t_{1} < ... < 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 a regular integral:
Important statements about path integrals are given by the Cauchy integral theorem and Cauchy's integral formula.
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