Line 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.
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 < ... < tn = 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
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Quantum mechanics
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