Defining a periodic function.

edit

Thank-you for the quick answer.

frac() gives a function which is periodic on positive reals. I just adapted your solution to get it periodic on \RR.

f = lambda x: 1 if (x - RR(x).floor()) < 1/2 else 0

I would have liked to be able to define a symbolic function though. Is it doable?

I was also trying to get a function whose plot is correct (without asking it to be pointwise) as it is the case for Piecewise().

Below are some of the things (not chronologically ordred though) I had tried (just for completness as I guess they are just full of beginer's classical mistakes).

f(x)=sum((-1)^k*unit_step(x-k),k,0,infinity)

which does not evaluate

f1(x)=0
f2(x)=1
h=Piecewise([[(-oo,0),f1],[(0,1),f2],[(1,2),f1],[(2,+oo),f1]],x)
h.plot()

returns ValueError: cannot convert float NaN to integer.

f(x)=sum(h(x-2*k),k,0,infinity)

returns ValueError: Value not defined outside of domain.

f(x)=sum(h(x-2*k),k,0,4)

also returns ValueError: Value not defined outside of domain.

I also tried to redefine unit_step:

def echelon_unite(x):
#
    if x<0:
        hres=0
    else:
        hres=1
    return hres

problem integral(echelon_unite(x),x,-10,3) returns 13

numerical integral returns a coherent result.

Other tentative with incoherent result (still with integrate and not numerical_integral)

  def Periodisation_int(f,a,b):
        x = var('x')
        h0(x) = (x-b)-(b-a)*floor((x-b)/(b-a))
        hres = compose(f,h0)
        return hres


sage: g=Periodisation_int(sin,0,1)
sage: integrate(g(x),x,0,2)
-cos(1) + 2
sage: integrate(g(x),x,0,1)
-cos(1) + 1
sage: integrate(g(x),x,1,2)
-1/2*cos(1) + 1

My guess is I was using integrate on inappropriate objects. I would still like to know how to define corresponding symbolic function (if it is possible).

Thanks again,

best regards.