Re: [Mathematica 6] Integrate strange result

*To*: mathgroup at smc.vnet.net*Subject*: [mg78379] Re: [Mathematica 6] Integrate strange result*From*: Bill Rowe <readnewsciv at sbcglobal.net>*Date*: Fri, 29 Jun 2007 05:52:50 -0400 (EDT)

On 6/28/07 at 4:29 AM, nma at 12000.org (Nasser Abbasi) wrote: >The problem is this: When I ask Mathematica to integrate something >involving some constant parameter, it gives an answer for some range >of the constant involved, say range A, and it says there is no >answer for the other range, say range B >But next, when I specify, using assumptions, that the constant is >now in range B, then Mathematica does given an answer to the >integral ! >Why does it in one case say there is no answer if the constant is in >range B, and in the second case it gives an answer when the constant >is in range B? >case 1 ======== >In[2]:= f = (1 + k*Sin[a]^2)^(1/2); >In[12]:= Assuming[Element[k, Reals], Integrate[f, {a, 0, 2*Pi}]] >Out[12]= If[k >= -1, 4*EllipticE[-k], Integrate[Sqrt[1 + >k*Sin[a]^2], {a, 0, 2*Pi}, Assumptions -> k < -1]] >notice in the above, it says for k<-1 there is NO answer No, it appears to me to be saying the answer 4*EllipticE[-k] is valid when k >= -1 and an answer wasn't found for k < -1. That isn't the same as no answer. But having pointed that out there does seem to be a problem here. If I use k = -2 then the integrand becomes In[33]:= f = (1 + k*Sin[a]^2)^(1/2); In[34]:= f /. k -> -2 Out[34]= Sqrt[1 - 2*Sin[a]^2] looking at the result of Plot[(1 - 2.*Sin[a]^2)^0.5, {a, 0, 2 Pi}, Ticks -> {Range[0, 2 Pi, Pi/4], Automatic}] it is apparent the result for Integrate[Sqrt[1 - 2*Sin[a]^2], {a,0, 2 Pi}] should be the same as 4 Integrate[Sqrt[1 - 2*Sin[a]^2], {a,0, Pi/4}]. Using the documentation center to get information for EllipticE I find this integral should be 4 EllipticE[Pi/4,2] Checking In[35]:= 4 EllipticE[\[Pi]/4, 2.] // Chop Out[35]= 2.39628 and In[36]:= 4*NIntegrate[(1 - 2.*Sin[a]^2)^0.5, {a, 0, Pi/4}] Out[36]= 2.39628 So, the answer you get below cannot be correct in general. >case 2 >======= >In[2]:= f = (1 + k*Sin[a]^2)^(1/2); >In[10]:= Assuming[Element[k, Reals] && k < -1, Integrate[f, {a, 0, >2*Pi}]] >Out[10]= 0 >Notice in the above, for k<-1 it gives an answer, which is zero. -- To reply via email subtract one hundred and four