average value of function

The set of the values of a real function f defined on an intervalMathworldPlanetmathPlanetmath[a,b]  is usually uncountable, and therefore for being able to speak of an average value of f in the sense of the average value

A.V.=a1+a2++ann=j=1najj=1n1 (1)

of a finite list  a1,a2,,an  of numbers, one has to replace the sums with integrals.  Thus one could define



A.V.(f):=1b-aabf(x)𝑑x. (2)

For example, the average value of x2 on the interval  [0, 1]  is 13 and the average value of sinx on the interval  [0,π]  is 2π.

The definition (2) may be extended to complex functions f on an arc γ of a rectifiable curve via the contour integral

A.V.(f):=1l(γ)γf(z)𝑑z (3)

where l(γ) is the length (http://planetmath.org/ArcLength) of the arc.  If especially γ is a closed curve in a simply connected domain where f is analyticPlanetmathPlanetmath, then the average value of f on γ 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 CountableMathworldPlanetmath
Related topic GaussMeanValueTheorem
Related topic Expectation
Related topic MeanSquareDeviation
Defines average value