rectification of antiperiodic function

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

Then the function $f_1$ defined by

$f_1(t):=\max \{f(t),\mathrm{\hspace{0.17em}0}\}=\{\begin{array}{cc}f(t)\hspace{1em}\text{for}f(t)>0,\hfill & \\ 0\hspace{1em}\hspace{1em} \text{for}f(t)\leqq 0\hfill & \end{array}$

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

$$ℒ\{f_1(t)\}=\frac{1}{1-e^{-ps}}F(s),$$

$$ℒ\{f_2(t)\}=\frac{1+e^{-ps}}{1-e^{-ps}}F(s).$$