# Defining a periodic function.

Hi,

I am a new sage user. I'd like to define simple periodic maps over \R which plot and integrate correctly (eg. a square signal (which of course is discontinuous but which I would still like to be able to plot in a way that makes it clear that the function is not multivalued at discontinuity points)). I tried different approaches none of which gave satisfactory results.

Any hint on how to do that nicely (or what would be the obstacles)?

Thank you

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( 2013-09-17 18:39:55 -0500 )edit

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The object you get out of Periodisation_int is not a symbolic function, but when you call it with x you get a symbolic expression anyway, so it doesn't really matter (and most symbolic machinery is for expressions anyway). If you really want a symbolic function you can do:

def per(f,a,b):
x = SR.var('x')
CSRx = CallableSymbolicExpressionRing( (x,))
t = f((x-b)-(b-a)*floor((x-b)/(b-a)))
h = CSRx(t)
return h


where the whole bit with CSRx is to actually get a symbolic function (something you can call). I'm not so sure that integrate knows what to do with this symbolic function, though.

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You should tell us about your different approaches, and what was wrong with them. I am not sure whether you ask about defining a function or plotting it. Here would be my direct lazy approach, without more informations about your needs. To define a periodic function, use the .frac() method of real numbers. In your case:

sage: f = lambda x: 1 if RR(x).frac() < 1/2 else 0
sage: plot(f, 0, 4)


If you do not like the vertical line at the discontinuities, you can look at the options of the plot() function:

sage: plot?


In your case, you can try:

sage: plot(f, 0, 4, plot_points='1000', linestyle='', marker='.')


For the integral, since the finction is defined point by points (not a symbolic function), you can do a numerical integration:

sage: numerical_integral(f, 0, 4)
(2.0, 2.2315482794965646e-14)

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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.

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