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major clean-up/rewrite of integrals of even/odd |
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* The [[derivative]] of an even function is odd.
* The derivative of an odd function is even.
* If an odd function is [[integral|integrable]] over a [[Interval (mathematics)|bounded symmetric interval]] <math>[-A,A]</math>, the integral over that interval is zero; that is<ref>{{cite web|url=http://mathworld.wolfram.com/OddFunction.html|title=Odd Function|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com}}</ref>
*:<math>\int_{-A}^{A} f(x)\,dx = 0</math>.
** This implies that the [[Cauchy principal value]] of an odd function over the entire real line is zero.
* If an even function is integrable over a bounded symmetric interval <math>[-A,A]</math>, the integral over that interval is twice the
*:<math>\int_{-A}^{A} f(x)\,dx = 2\int_{0}^{A} f(x)\,dx</math>.
** This property is also true for the [[improper integral]] when <math>A = \infty</math>, provided the intergral from 0 to <math>\infty</math> converges.
===Series===
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