# 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

$A.V.={\displaystyle \frac{{a}_{1}+{a}_{2}+\mathrm{\dots}+{a}_{n}}{n}}={\displaystyle \frac{{\sum}_{j=1}^{n}{a}_{j}}{{\sum}_{j=1}^{n}1}}$ | (1) |

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

$$A.V.(f):=\frac{{\int}_{a}^{b}f(x)\mathit{d}x}{{\int}_{a}^{b}1\mathit{d}x},$$ |

i.e.

$A.V.(f):={\displaystyle \frac{1}{b-a}}{\displaystyle {\int}_{a}^{b}}f(x)\mathit{d}x.$ | (2) |

For example, the average value of ${x}^{2}$ on the interval $[0,\mathrm{\hspace{0.17em}1}]$ is $\frac{1}{3}$ and the average value of $\mathrm{sin}x$
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

$A.V.(f):={\displaystyle \frac{1}{l(\gamma )}}{\displaystyle {\int}_{\gamma}}f(z)\mathit{d}z$ | (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 |