# rectification of antiperiodic function

Let the positive number $p$ be the antiperiod of the real function $f$ and

 $f(t)\geqq 0\quad\mbox{for}\;\;0

Then the function $f_{1}$ defined by

 $\displaystyle f_{1}(t)\;:=\;\max\{f(t),\,0\}\;=\;\begin{cases}f(t)\quad\mbox{% for}\;\;f(t)>0,\\ 0\qquad\,\mbox{for}\;\;f(t)\leqq 0\end{cases}$

is the half-wave rectification of $f$ and the function $f_{2}$ defined by

 $f_{2}(t)\;:=\;|f(t)|$

is the full-wave rectification of $f$.  They are periodic (http://planetmath.org/PeriodicFunctions), the former with period (http://planetmath.org/PeriodicFunctions) $2p$ and the latter with $p$.

The Laplace transforms are

 $\mathcal{L}\{f_{1}(t)\}\;=\;\frac{1}{1\!-\!e^{-ps}}F(s),$
 $\mathcal{L}\{f_{2}(t)\}\;=\;\frac{1\!+\!e^{-ps}}{1\!-\!e^{-ps}}F(s).$
 Title rectification of antiperiodic function Canonical name RectificationOfAntiperiodicFunction Date of creation 2013-03-22 18:58:19 Last modified on 2013-03-22 18:58:19 Owner pahio (2872) Last modified by pahio (2872) Numerical id 6 Author pahio (2872) Entry type Definition Classification msc 44A10 Classification msc 26A99 Synonym rectification Related topic MaximalNumber Related topic LaplaceTransformOfPeriodicFunctions Related topic MinimalAndMaximalNumber Defines half-wave rectification Defines full-wave rectification