RE: Have I found a bug?

*To*: mathgroup at smc.vnet.net*Subject*: [mg84225] RE: [mg84209] Have I found a bug?*From*: "Jaccard Florian" <Florian.Jaccard at he-arc.ch>*Date*: Fri, 14 Dec 2007 07:03:36 -0500 (EST)*References*: <200712140855.DAA07725@smc.vnet.net>

Hello Louise, Why do you expect 1/Sqrt[1-x^2] ? If I look into the details : In[18]:= g[x_] := 1 - x^2 h[x_] := -x Exp[-Integrate[(D[g[x], x] - h[x])/g[x], x]] Out[20]= 1/Sqrt[x^2 - 1] In[21]:= f[x_] = (D[g[x], x] - h[x])/g[x] Out[21]= -(x/(1 - x^2)) In[22]:= k[x_] = Integrate[f[x], x] Out[22]= (1/2)*Log[x^2 - 1] In[23]:= E^(-k[x]) Out[23]= 1/Sqrt[x^2 - 1] I must say that I agree with Mathematica. I suppose your reason to think about a bug is the following : Integrating k[x] by hand, you obtain : (1/2)*Log[Abs[1-x^2]] , which is = (1/2)Log[1-x^2] if -1<x<1. So, if -1<x<1, you obtain as final result : E^(-(1/2)Log[1-x^2]]=1/Sqrt[1-x^2] Is it so? Well, you have not to forget that if Abs[x]>1, then 1/2)*Log[Abs[1-x^2]]=(1/2)*Log[x^2-1] and you would so obtain the same = result as Mathematica. The reason is very simple : "Integrate" gives a primitive which is = defined only modulo an arbitrary constant. Mathematica works in the = complex space, so you can for example choose the integrating constant = I*Pi/2. Integrating k[x], you would, with -1<x<1, have obtained : (1/2)Log[Abs[1-x^2]]+I*Pi/2 = (1/2)Log[1-x^2]+I*Pi/2 And so, your final result would be : E^(-((1/2)Log[1-x^2]+I*Pi/2))=1/Sqrt[1-x^2]*E^(-I*Pi/2)=1/Sqrt[1-x^2]= *(-I) But as -1<x<1, Sqrt[1-x^2]=Sqrt[-1*(x^2-1)]=-I*Sqrt[x^2-1] (because if a and b are negative, Sqrt[a*b]=-Sqrt[a]*Sqrt[b] ) So you finally obtain : 1/(-I*Sqrt[x^2-1])*(-I)=1/Sqrt[x^2-1] Which is also the same result as Mathematica. So I definitively think Mathematica is doing very well! Regards Florian Jaccard -----Message d'origine----- De=A0: Louise Hoffman [mailto:louise.hoffman at gmail.com] Envoy=E9=A0: vendredi, 14. d=E9cembre 2007 09:55 =C0=A0: mathgroup at smc.vnet.net Objet=A0: [mg84209] Have I found a bug? Dear readers, When I calc. this in Mathematica 5.2 g[x_] := 1 - x^2 h[x_] := -x Exp[-Integrate[ (D[g[x], x] - h[x])/g[x], x ] ] it returns 1/Sqrt[-1+x^2] where I would expect 1/Sqrt[1-x^2] Have I found a bug, or have I made a mistake? Lots of love, Louise

**References**:**Have I found a bug?***From:*Louise Hoffman <louise.hoffman@gmail.com>