Path integral

This is not about "path integrals" in the sense that means that which was studied by Richard Feynman.

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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

<math>\int_\gamma f(z)\,dz</math>

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

<math>\sum_{1 \le k \le n} f\left( \;\gamma(t_k)\;\right) \left[ \; \gamma(t_k) - \gamma(t_{k-1}) \; \right]</math>

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:

<math>\int_\gamma f(z)\,dz \int_a^b

f( \;\gamma(t)\; ) \; \gamma^\prime(t) dt</math>

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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