Re: How to convert a HeavisideTheta to a PieceWise function

*To*: mathgroup at smc.vnet.net*Subject*: [mg93797] Re: How to convert a HeavisideTheta to a PieceWise function*From*: Nikolaus Rath <Nikolaus at rath.org>*Date*: Wed, 26 Nov 2008 05:12:23 -0500 (EST)*References*: <gggqf5$317$1@smc.vnet.net>

"wxziwyz at 126.com" <wangxz1983 at gmail.com> writes: > For example: > we have f(x) = (x + (-2 + x) HeavisideTheta[-2 + x] - 2 (-1 + x) > HeavisideTheta[-1 + x]) HeavisideTheta[x] > > apparently, this is a piecewise function. > > My problem is, How to transfer it into a Piecewise expression, i.e. > > f(x) = Piecewise[{{0, x < 0}, {x, 0 < x <= 1}, {2- x, 1 < x <= 2}= , {0, > x > 2}}] > > Is there any function in Mathematica can do this? I don't know any such function. But it should be possible to do it manually. The following seems to work at least for simple cases: f[x_] := HeavisideTheta[x - 2] x^2 + HeavisideTheta[x + 2] x; thetas = Extract[f[x], Position[f[x], HeavisideTheta[_]]]; cases = Solve[# == 0, x] & /@ thetas[[All, 1]]; boundaries = (x /. #) & /@ Flatten[cases]; boundaries = {-\[Infinity], Sequence @@ Sort[boundaries], \[Infinity]}; cases2 = Table[ boundaries[[i]] < x < boundaries[[i + 1]], {i, Length[boundaries] - 1}] Piecewise [ {Simplify[f[x], Assumptions -> {#}], # } & /@ cases2] It first extracts the arguments of all HeavisideTheta functions. Then it solves them for x to determine the positions at which the Heaviside function changes from 0 to 1. It appends -/+ Infinity to this list of boundaries and constructs a list of cases. Finally, Simplify is used to get rid of the Heaviside's inside each region. HTH, -Nikolaus -- =C2=BBIt is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.=C2=AB -J.H. Hardy PGP fingerprint: 5B93 61F8 4EA2 E279 ABF6 02CF A9AD B7F8 AE4E 425C