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| − | The same as the [[Variation of a function|variation of a function]] of one variable. The total variation of a real-valued function is the sum of its positive variation (cf. [[Positive variation of a function|Positive variation of a function]]) and negative variation (cf. [[Negative variation of a function|Negative variation of a function]]).
| + | #REDIRECT[[Variation of a function]] |
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| − | ====Comments====
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934001.png" /> is a complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934002.png" />, then its total variation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934003.png" /> is the number
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934004.png" /></td> </tr></table>
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934005.png" /> is also continuous, then this number is the same as the length of the arc in the complex plane that is parametrized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934006.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934007.png" /> is absolutely continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934008.png" />, then
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093400/t0934009.png" /></td> </tr></table>
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)</TD></TR></table>
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Latest revision as of 14:36, 18 September 2012
How to Cite This Entry:
Total variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_variation_of_a_function&oldid=17306
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article