simplifications
- To: mathgroup at smc.vnet.net
- Subject: [mg74517] simplifications
- From: "dimitris" <dimmechan at yahoo.com>
- Date: Fri, 23 Mar 2007 19:17:32 -0500 (EST)
Hello. Below I use Mathematica to get some definite integrals along with simplifications of the results. ln[9]:= f1 = FullSimplify[Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, 0, Pi}]] Out[9]= (1/8)*Pi*(2*Pi + 8*I*ArcSinh[1] + Log[((2 - Sqrt[2])^(9*I)*(2 + Sqrt[2])^(7*I))/(10 - 7*Sqrt[2])^(3*I)] + Log[(6 + 4*Sqrt[2])^(-6*I)] - I*Log[10 + 7*Sqrt[2]]) In[10]:= f2 = FullSimplify[Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, -2*Pi, 2*Pi}]] Out[10]= -Pi^2 In[11]:= f3 = FullSimplify[Tr[(Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, #1[[1]], #1[[2]]}] & ) /@ Partition[Range[0, Pi, Pi/2], 2, 1]]] Out[11]= Pi^2/4 In[12]:= f4 = FullSimplify[Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, 0, Pi/2, Pi}]] Out[12]= Pi^2/4 In[13]:= f5 = FullSimplify[Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, 0, 5*Pi}]] Out[13]= (5*Pi^2)/4 In[14]:= f6 = FullSimplify[Integrate[x*(Sin[x]/(1 + Cos[x]^2)), {x, -Pi, Pi}]] Out[14]= (1/4)*Pi*(2*Pi + 8*I*ArcSinh[1] + Log[((2 - Sqrt[2])^(9*I)*(2 + Sqrt[2])^(7*I))/(10 - 7*Sqrt[2])^(3*I)] + Log[(6 + 4*Sqrt[2])^(-6*I)] - I*Log[10 + 7*Sqrt[2]]) No matter what I try I couldn't simplify f1 and f6 directly to Pi^2/4 and Pi^2/2. The only thing I could show (good news of course but this is not my point!) was In[20]:= Block[{Message}, FullSimplify[ (1/8)*Pi*(2*Pi + 8*I*ArcSinh[1] + Log[((2 - Sqrt[2])^(9*I)*(2 + Sqrt[2])^(7*I))/(10 - 7*Sqrt[2])^(3*I)] + Log[(6 + 4*Sqrt[2])^(-6*I)] - I*Log[10 + 7*Sqrt[2]]) == Pi^2/4, ComplexityFunction -> (Count[{#1}, _Log | _ArcSinh, Infinity] & )]] Out[20]= True In[23]:= Block[{Message}, FullSimplify[ (1/4)*Pi*(2*Pi + 8*I*ArcSinh[1] + Log[((2 - Sqrt[2])^(9*I)*(2 + Sqrt[2])^(7*I))/(10 - 7*Sqrt[2])^(3*I)] + Log[(6 + 4*Sqrt[2])^(-6*I)] - I*Log[10 + 7*Sqrt[2]]) == Pi^2/2, ComplexityFunction -> (Count[{#1}, _Log | _ArcSinh, Infinity] & )]] Out[43]= True Any ideas??? PS Can somebody guess why we f1 and f6 do not simplify directly as the other fi?