average value of function
The set of the values of a real function defined on an interval is usually uncountable, and therefore for being able to speak of an average value of in the sense of the average value
(1) |
of a finite list of numbers, one has to replace the sums with integrals. Thus one could define
i.e.
(2) |
For example, the average value of on the interval is and the average value of
on the interval is .
The definition (2) may be extended to complex functions on an arc of a rectifiable curve via the contour integral
(3) |
where is the length (http://planetmath.org/ArcLength) of the arc. If especially is a closed curve in a simply connected domain where is analytic, then the average value of on is always 0, as the Cauchy integral theorem implies.
Title | average value of function |
Canonical name | AverageValueOfFunction |
Date of creation | 2013-03-22 19:01:54 |
Last modified on | 2013-03-22 19:01:54 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 26D15 |
Classification | msc 11-00 |
Related topic | ArithmeticMean |
Related topic | Mean3 |
Related topic | Countable |
Related topic | GaussMeanValueTheorem |
Related topic | Expectation |
Related topic | MeanSquareDeviation |
Defines | average value |