Piecewise function with integer conditions

Question

I'm trying to define (and plot) a piecewise function with the following structure:

$$f(x) = \begin{cases} g(x) & nT \leq x \leq \frac{2n+1}{2}T\\ -g(x) & \frac{2n+1}{2}T \leq x \leq (n+1)T \end{cases}$$

where n is an integer, n >= 0 and T is a constant.

The idea is that it cycles between positive and negative after a full period is complete.

My attempt was:

f[x_] := Piecewise[{
     {g[x], (n T <= x <= ((2 n + 1)/2) T) && Element[n, Integers] && (n >= 0)},
     {-g[x], (((2 n + 1)/2) T <= x <= (n + 1) T) && Element[n, Integers] && (n >= 0)}}]

But when I try a test value for x in f[x] it shows n in the conditions for the output, so it seems to not get past the conditions. I believe plotting fails for this reason too.

I have never tried to use Integers in Mathematica.

Answer 1

Your problem is that you haven't told Piecewise that n can be arbitrary, so Mathematica thinks it's constant. You'd be better off defining n as Floor[x/T] for the first comparison, rather than asking Mathematica to find an n such that the definition holds (or prove that n doesn't exist).

It looks like you could simplify the second condition to True, by the way: Piecewise checks the pieces in order, so there's no need to give a complicated final condition.

Answer 2

A way following your line of thought:

T = Pi;
g[x_] := Sin@x
f[x_?NumericQ] := Piecewise[
     {{g[x] , Quiet@Resolve@Exists[n, n ∈ Integers && (n + 1/2)T <= x <= (n + 1)T]}},
      -g[x]]
Plot[f[x], {x, 0, 10}]

I would do it this way instead:

Plot[-g[x] SquareWave[x/T], {x, 0, 10}]

And from there you can get an equivalent Piecewise form:

PiecewiseExpand[-g[x] SquareWave[x/T]]

$$\begin{cases} -g(x) & 0\leq (\frac{x}{T} \bmod 1)<\frac{1}{2} \\ g(x) & \text{True} \end{cases}$$