Definition:Cauchy Integral

Definition

Let $f$ be a continuous real function.

Let $\closedint a b$ be a closed real interval such that $a < b$.

Let $P = \set {a, x_1, x_2, \ldots, x_{n - 1}, b}$ be a normal subdivision of $\closedint a b$ such that $x_k - x_{k - 1} = \delta x$.

The Cauchy integral of $f$ from $a$ to $b$ is defined as:

$\ds A = \lim_{\delta x \mathop \to 0} \sum \map f x \rdelta x$

Also see

• Results about Cauchy integrals can be found here.

Source of Name

This entry was named for Augustin Louis Cauchy.

Historical Note

The method of forming a definite integral known as a Cauchy integral was first put forward by Augustin Louis Cauchy.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy integral
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integration
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy integral
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integration