# average value of function

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

 $\displaystyle A.V.\;=\;\frac{a_{1}\!+\!a_{2}\!+\ldots+\!a_{n}}{n}\;=\;\frac{% \sum_{j=1}^{n}a_{j}}{\sum_{j=1}^{n}1}$ (1)

of a finite list  $a_{1},\,a_{2},\,\ldots,\,a_{n}$  of numbers, one has to replace the sums with integrals.  Thus one could define

 $A.V.(f)\;:=\;\frac{\int_{a}^{b}\!f(x)\,dx}{\int_{a}^{b}\!1\,dx},$

i.e.

 $\displaystyle A.V.(f)\;:=\;\frac{1}{b\!-\!a}\int_{a}^{b}\!f(x)\,dx.$ (2)

For example, the average value of $x^{2}$ on the interval  $[0,\,1]$  is $\frac{1}{3}$ and the average value of $\sin{x}$ on the interval  $[0,\,\pi]$  is $\frac{2}{\pi}$.

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

 $\displaystyle A.V.(f)\;:=\;\frac{1}{l(\gamma)}\int_{\gamma}\!f(z)\,dz$ (3)

where $l(\gamma)$ is the length (http://planetmath.org/ArcLength) of the arc.  If especially $\gamma$ is a closed curve in a simply connected domain where $f$ is analytic, then the average value of $f$ on $\gamma$ 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