# Cauchy integral

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A Cauchy integral is a definite integral of a continuous function of one real variable. Let $f (x)$ be a continuous function on an interval $[a, b]$ and let $a = x _ {0} < \dots < x _ {i - 1 } < x _ {i} < \dots < x _ {n} = b$, $\Delta x _ {i} = x _ {i} - x _ {i - 1 }$, $i = 1, \dots, n$. The limit

$$\lim\limits _ {\max \Delta x _ {i} \rightarrow 0 } \ \sum _ {i = 1 } ^ { n } f (x _ {i - 1 } ) \Delta x _ {i}$$

is called the definite integral in Cauchy's sense of $f (x)$ over $[a, b]$ and is denoted by

$$\int\limits _ { a } ^ { b } f (x) dx.$$

The Cauchy integral is a particular case of the Riemann integral. The definition is due to A.L. Cauchy .

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How to Cite This Entry:
Cauchy integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_integral&oldid=52051
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article