Re: Integrate with If and Which

*To*: mathgroup at smc.vnet.net*Subject*: [mg20203] Re: [mg20167] Integrate with If and Which*From*: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>*Date*: Wed, 6 Oct 1999 21:06:24 -0400*Sender*: owner-wri-mathgroup at wolfram.com

There is obviously no "logical" principle involved here. Since I have no access to the Mathematica code I can only speculate about the reasons, but still I can think of one point which throws some light on this (at least on the "why If but not Which?" issue). Note that Integrate works only with the simplest If construction. Thus: In[1]:= Integrate[If[x < 1, 1, 2], {x, 0, 2}] Out[1]= 3 but In[2]:= Integrate[If[x < 1, 1, If[x < 2, 2, 3]], {x, 0, 3}] Out[2]= Integrate[If[x < 1, 1, If[x < 2, 2, 3]], {x, 0, 3}] Both of these can be written in terms of Which, but the first statement corresponds to a rather trivial form of Which, namely Which[x<1,1,True,2]. A general Which corresponds to a nested sequence of Ifs and as we have seen for such a sequence Integrate also does not work. One could certainly make Integrate work with simple Which statements, e.g. in the following way: In[3]:= Unprotect[Integrate]; In[4]:= Integrate /: Integrate[Which[a_, b_, True, d_], {l__}] := Integrate[If[a, b, d], {l}] In[5]:= Protect[Integrate]; Now indeed we have In[6]:= Integrate[Which[x < 1, 1, True, 2], {x, 0, 2}] Out[6]= 3 but you can't say this is really much progress, since anyway Which is not intended for such simple single clause statements. So really your question should be not "Why If but not Which?" but "Why no nested If's?". If we could Integrate nested Ifs we could extend this to Which. The answer to this question I really do not know, although I have made a brief attempt to find a recursive way to integrate nested Ifs and I can say at least that it is seems hard to do in reasonable generality. -- Andrzej Kozlowski Toyama International University JAPAN http://sigma.tuins.ac.jp http://eri2.tuins.ac.jp ---------- >From: "L. Dwynn Lafleur" <lafleur at usl.edu> To: mathgroup at smc.vnet.net >To: mathgroup at smc.vnet.net >Subject: [mg20203] [mg20167] Integrate with If and Which >Date: Mon, Oct 4, 1999, 10:07 > > It has been pointed out before in this newsgroup that Mathematica integrates > some conditional functions but not others. For example, consider the > following text translation of a notebook from version 4: > > In[1]:= f[u_] := If[u < 0, u, u^2]; > g[u_] := Which[u < 0, u, u >= 0, u^2]; > > In[3]:= Integrate[f[u], {u, -1, 1}] > > Out[3]= -(1/6) > > In[4]:= Integrate[g[u], {u, -1, 1}] > > Out[4]= Integrate[Which[u < 0, u, u >= 0, u^2], {u, -1, 1}] > > Functions f[u] and g[u] are mathematically identical integrands, but > Mathematica integrates only the former. You can force numerical evaluation > of the latter by wrapping it in N[]. > > My question is, "What is the fundamental difference between If and Which > that makes Mathematica treat them differently?" As I said above, this > Mathematica "feature" has been pointed out before and ways to avoid it have > been described, but I don't recall a post giving the reason for the > behavior. I guess I am just curious to know if there is a logical principle > involved. > > > Dwynn > > -- > > ========================================== > L. Dwynn Lafleur > Professor of Physics > University of Louisiana at Lafayette > lafleur at usl.edu > ========================================== > > > > > >