Path integral
This isn't about "path integrals" in the sense that means that which was studied by Richard Feynman.
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 < ... < tn = 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
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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