  ## Encyclopedia > Path integral

Article Content

# Path integral

This is not 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 : UC is a function. Then the path integral

$\int_\gamma f(z)\,dz$

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

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

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:

$\int_\gamma f(z)\,dz \int_a^b f( \;\gamma(t)\; ) \; \gamma^\prime(t) dt$

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

fill in

fill in

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Thomas a Kempis ... of Christ," was born at Kempen[?], Germany (40 miles northwest of Cologne) in 1380 and died near Zwolle (52 miles east-north-east of Amsterdam) in 1471. His paternal ...  