Continuous Integrals for Piecewise Continuous Functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg17267] Continuous Integrals for Piecewise Continuous Functions*From*: Jack Goldberg <jackgold at math.lsa.umich.edu>*Date*: Fri, 30 Apr 1999 02:34:47 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Hi Group, After trying many different variants, I have finally settled on a scheme that provides a simple answer to the question of integrating piecewise continuous functions (abbreviated PC from now on :-). The key idea is this: The rhs of the formula, Integrate[f_.*Sign[x_-a_.].x_] = Integrate[f,{x,a,x}]*Sign[x-a] is continuous for all x and its derivative exists and is f*Sign[x-a] at all x except possibly at x=a. Since f/2*(Sign[x-a] - Sign[x-b]) is the PC function which vanishes outside of [a,b] and is f in (a,b), sums of this type can be used to construct any PC function and the integration formula then provides the required continuous anti-derivative. (Sums of continuous functions are continuous). Here is the code without bells and whistles. In[1]: MakePC[G:{_,{x_,_,_}}..] := Apply[Plus, List[G] /. ({f_,{x,{a_,b_}}->f/2*Sign[x-a]-f/2*Sign[x-b]) ( A typical argument of MakePC is {expr,{x,a,b}} which is meant to denote a function expr in (a,b), zero outside of [a,b]. The code can be understood easily enough. What happens here is this: The sequence named G becomes a list and each entry in this list is converted into the appropriate difference of Sign functions. The "Apply" converts the list to a sum). Some nice examples. Plotting helps. In[2]:= SquareWave[x_] := MakePC[{1,{x,0,1}] Out[2]= -1/2 Sign[-1+x]+Sign[x]/2 In[3]:= Tent[x_] := MakePC[{1+x,{x,-1,0}},{1-x,{x,0,1}}] Out[3]= -1/2(1-x) Sign[-1+x] +1/2(1-x)Sign[x]-1/2(1+x)Sign[x] + 1/2(1+x)Sign[1+x] I have written code that will clean up the Out[3], but I don't want to clutter this post but see summary below). In[4]:= f[x_] := MakePC[{-1,{x,-Infinity,0}},{1,{x,0,Infinity}}] Out[4]= Sign[x] (Sign handles Infinities well). Now the integration. In[5]:= Unprotect[Integrate]; In[6]:= Integrate[f_.*Sign[x_+a_.],x_] := Integrate[f,{x,-a,x}*Sign[x+a] (Note the -a in the rhs. This is due to the argument x+a of Sign.) In[7]:= Integrate[h_+f_.*g_Sign,x_] := Integrate[h,x]+Integrate[f*g,x] In[8]:= Protect[Integrate] Since the result of MakePC is a sum of the form f*Sign[...], In[7] and the definition In[6] should take care of all cases of MakePC that are "well-formulated". Some final comments: There are at least 6 extensions (improvements?) that can be made to the code. I have done them all and will send them to anyone interested. (1) If MakePC has a single argument the outer {} are not necessary and we could add this as a special case. (2) An option to MakePC might be to plot the graph since I have found it somewhat tedious trying to see if I really did get the right function constructed. (3) A list of rules (I called them SignRules) which enables us to manipulate more complicated functions of Sign such as Tent[x]^2*SquareWave[x]- Sign[x] I can't think of any use for such a function; but since I knew it was PC, I challanged myself to see if I could simplify it by using a collection of rules. (4) Clean up the output of MakePC (5) Construct the code to handle definite integrals. (6) Error catching steps. I worked alone on this. With no feedback to catch my often silly programming errors, I would be very appreciative of any remarks. My ego is not involved in this. I just hope to stimulate both the people at Mathematica (If I can do it, how come its not available?) and those generous experts who have contributed so much to this group. Jack Goldberg Univ. of Mich Math

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