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Re: Piecewise functions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51581] Re: [mg51553] Piecewise functions
*From*: "David Park" <djmp at earthlink.net>
*Date*: Sat, 23 Oct 2004 00:22:36 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Luca,
Not totally clear. But here are two ways to define the function.
Clear[y]
y[x_] /; x > 3 := x
y[x_] /; -1 < x <= 3 := -x
y[x_] := 1
Plot[y[x], {x, -2, 4}];
But this method does not allow you to correctly calculate
y'[x]
0
So, UnitStep is usually the better way.
Clear[y]
y[x_] := 1(1 - UnitStep[x + 1]) - x(UnitStep[x + 1] - UnitStep[x - 3]) +
x UnitStep[x - 3]
This can also be written as
Clear[y]
y[x_] := 1(UnitStep[x + Infinity] - UnitStep[x + 1]) -
x(UnitStep[x + 1] - UnitStep[x - 3]) + x UnitStep[x - 3]
which has the form of a sum of (value expression)(turn on point - turn off
point)
Plot[y[x], {x, -2, 4}];
Now, you could integrate the function...
yintegral[x_] = Integrate[y[t], {t, -2, x}]
2 + x - 9*UnitStep[-3 + x] + x^2*UnitStep[-3 + x] -
(1/2)*UnitStep[1 + x] - x*UnitStep[1 + x] -
(1/2)*x^2*UnitStep[1 + x]
Plot[yintegral[x], {x, -2, 4}];
or differentiate it.
y'[x]
x DiracDelta[-3 + x] - DiracDelta[1 + x] -
x (-DiracDelta[-3 + x] + DiracDelta[1 + x]) + 2 UnitStep[-3 + x] -
UnitStep[1 + x]
But the plot usually doesn't catch the deltas.
Plot[y'[x], {x, -2, 4}, PlotRange -> All];
With DrawGraphics, from my web site, it's easy to make somewhat better
plots.
Needs["DrawGraphics`DrawingMaster`"]
DrawUnitStep eliminates the jump lines at the discontinuities.
Draw2D[
{DrawUnitStep[y[x], {x, -2, 4}]},
Frame -> True,
PlotLabel -> HoldForm[y[x]],
Background -> Linen,
ImageSize -> 400];
We can even get in the DiracDeltas by converting them to Arrows with the
following routine. Notice that the second delta is actually zero.
Cases[Collect[y'[x], DiracDelta[_]], a_ DiracDelta[x + b_]]
diracs = % /.
a_ DiracDelta[x + b_] :>
Arrow[{-b, 0}, {-b, (a /. x -> -b)}, HeadCenter -> 1/2]
Draw2D[
{DrawUnitStep[y'[x], {x, -2, 4}],
Red, diracs},
Frame -> True,
PlotRange -> All,
Background -> Linen,
PlotLabel -> HoldForm[y'[x]],
ImageSize -> 400];
David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/
From: Luca [mailto:luca at nospam.it]
To: mathgroup at smc.vnet.net
Hi all. I'm studying for the exam of signals and systems and I was
trying to plot some kind of functions I transformed for exercise. So, I
need to plot piecewise functions like:
y(x) = x if x > 3
y(x) = -x if -1 < x < 3
y(x) = 1 else
(should have been a system).
I found out in the guide the chapter about this, and I learned that it
is possible with the function UnitStep, which I know. Anyway, I found
it difficult to determine the equation of the function using this
method. Is it possible to do it simply writing everything like I did
before, more or less? i.e. without having to determine the equation
with the UnitStep function.
Hope I've been clear enought. Many thanks.
Luca
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