Programming Options for Power Expand

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg802] Programming Options for Power Expand*From*: Jack Goldberg <jackgold at math.lsa.umich.edu>*Date*: Fri, 21 Apr 1995 01:41:22 -0400

I have received a number (1) of requests for my solution to my own problem: Enhance PowerExpand so that among other things it simplifies ArcTan[Tan[x]] = x. The real question is more general: Should (and can!) one add options to a built in command? I do not know the answer to whether one should. I am interested in knowing what is good Mma programming practice in this regard. Is what I present next poor programming practice? Why? (I am most interested in the style, less in how to improve my code.) Step(1): Define rules. I list a few: In[1]: ArcRules = { ArcTan[ Tan[x_] ] :> x, ArcTan[ Cot[x_] ] :> Pi/2 - x, .... ArcCsc[ Sec[x_] ] :> Pi/2 - z}: The .... refers to all those combinations ArcYyy[ [Yyy[x_] ] :> x which I left out to save space. Step(2) In[2}: ArcRules = Dispatch[ArcRules]; ( *Don't ask me to justify this step. I just lifted it out of Roman Maeder's package "Algebra`Trigonometry`". *) Step (3) In[3]: Unprotect[PowerExpand]; In[4]: PowerExpand[expr_,InverseTrig->True] := Module[ {fnt}, fnt = Simplify[ PowerExpand[#//.ArcRules] ]&; FixedPoint[ fnt,expr ] ] In[5]: Protect[PowerExpand]; Some trial examples: ex1 = 1 + Log[ArcTan[Tan[Exp[x]]]]; ex2 = 1 + ArcTan[Log[Exp[Tan[x]]]]; ex3 = ArcSin[Sqrt[1-Cos[x]^2]]; PowerExpand without the "option" leave these expressions unaltered. PowerExpand[ex1,InverseTrig->True] simplifies to 1+x. The others work similarly. I think that I have cheated here. InverseTrig->True acts like an option but it strikes me that it is in fact a second argument to PowerExpand. But perhaps that's all options are anyway. Any discussion on the matters raised by this post are appreciated. I learn even from those I disagree with! Thanks all. Jack P.S. I hope I made it clear that other rules such as ArcTanh[ Tanh[x_] ] :> x can be added to the list in ArcRules at your whim.