Integrate Bug?

*To*: mathgroup at smc.vnet.net*Subject*: [mg13123] Integrate Bug?*From*: Ed Hall <teh1m at virginia.edu>*Date*: Tue, 7 Jul 1998 03:44:26 -0400*Organization*: University of Virginia*Sender*: owner-wri-mathgroup at wolfram.com

Folks, The following integration problem appears to be a bug in Mathematica's Integrate function and Wolfram's technical support has been unable to help so far. I was hoping someone reading this newsgroup might be able to come up with a solution. I can create the expression below (from the Peng-Robinson equation of state) which is real, differentiate w.r.t the variable W, followed by indefinite integration w.r.t. to W to get a complex result rather than the original real expression. I've tried adding Im[W]==0 assumption to the Integrate command w/o success. Composing the expression to be differentiated. In[5]:= V = 1/W; d = Sqrt[u^2*Nt^2*b^2 - 4*w*b^2*Nt^2]; F = -(Nt*Log[(V - Nt*b)/V]) + (a*Nt^2*Log[(2*V + u*Nt*b - d)/ (2*V + u*Nt*b + d)])/(R*T*d) Out[6]= (a*Nt^2*Log[(b*Nt*u - Sqrt[b^2*Nt^2*u^2 - 4*b^2*Nt^2*w] + 2/W)/ (b*Nt*u + Sqrt[b^2*Nt^2*u^2 - 4*b^2*Nt^2*w] + 2/W)])/ (R*T*Sqrt[b^2*Nt^2*u^2 - 4*b^2*Nt^2*w]) - Nt*Log[(-b*Nt + 1/W)*W] In[7]:= F = FullSimplify[F] Out[7]= 1/(R*T)*Nt*(-R*T*Log[1 - b*Nt*W] + (a*Nt*Log[(2 + b*Nt*u*W - Sqrt[b^2*Nt^2* (u^2 - 4*w)]*W)/ (2 + b*Nt*u*W + Sqrt[b^2*Nt^2* (u^2 - 4*w)]*W)])/ Sqrt[b^2*Nt^2*(u^2 - 4*w)]) Taking partial derivative w.r.t. W In[8]:= test = D[F, W] Out[8]= 1/(R*T)*Nt*((b*Nt*R*T)/(1 - b*Nt*W) + (a*Nt*(2 + b*Nt*u*W + Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W)* (-(((b*Nt*u + Sqrt[b^2*Nt^2*(u^2 - 4*w)])* (2 + b*Nt*u*W - Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W))/ (2 + b*Nt*u*W + Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W)^2) + (b*Nt*u - Sqrt[b^2*Nt^2*(u^2 - 4*w)])/ (2 + b*Nt*u*W + Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W)))/ (Sqrt[b^2*Nt^2*(u^2 - 4*w)]* (2 + b*Nt*u*W - Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W))) Performing indefinite integration on result of differentiation. In[9]:= F1 = FullSimplify[Integrate[test, W]] Out[9]= 1/(R*T)*Nt*(-R*T*Log[-1 + b*Nt*W] + (a*Nt*(Log[-2 - b*Nt*u*W + Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W] - Log[2 + b*Nt*u*W + Sqrt[b^2*Nt^2*(u^2 - 4*w)]*W]))/ Sqrt[b^2*Nt^2*(u^2 - 4*w)]) Substituting numeric values into the original expression F (before differentiation and integration). In[10]:= N[F /. {b -> 1, Nt -> 2, u -> 2, w -> -1, W -> 0.25, a -> 1, R -> 1, T -> 1}] Out[10]= 0.662394 Substituting numeric values into the expression F1 after differentiation and integration. In[11]:= N[F1 /. {b -> 1, Nt -> 2, u -> 2, w -> -1, W -> 0.25, a -> 1, R -> 1, T -> 1}] Out[11]= 0.662394 - 4.06174 I The result F1 of the integration is complex whereas the original expression F before differention is real. How can I insure F1 will be real and equal to F? Thanks in advance, Ed ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ed Hall Research Computing Support edhall at virginia.edu Information Technology and Communication 804-924-0620 The University Virginia ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~