Re: Definite Integration in Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg74537] Re: Definite Integration in Mathematica*From*: Peter Pein <petsie at dordos.net>*Date*: Sat, 24 Mar 2007 05:25:05 -0500 (EST)*References*: <etqo3f$10i$1@smc.vnet.net> <ett6i6$gd2$1@smc.vnet.net> <eu1qr3$ih3$1@smc.vnet.net>

dh schrieb: > Hello Dimitris, > > Yes, you are right, Mathematica chooses an antiderivative function that > > has branch cuts at approx. +/-2. However, this is not the only possible > > antiderivative, there are others with different branch cuts. But it is > > not too easy to find them. > > Daniel > But it is quite easy to do this in Mathematica: In[1]:= << "DiscreteMath`" In[2]:= f[x_] = (4 + 2*x + x^2) / (17 + 2*x - 7*x^2 + x^4); In[3]:= F[x_] = Simplify[Assuming[k >= 0 && k \[Element] Integers, PowerSum[Simplify[ToRadicals[(x/(k + 1))*SeriesTerm[f[x], {x, 0, k}]]], {x, k}]]] Out[3]= (-(1/2))*I*(Log[34 + (1 + 4*I)*x - Sqrt[257 - 60*I]*x] + Log[34 + ((1 + 4*I) + Sqrt[257 - 60*I])*x] - Log[34 - ((-1 + 4*I) + Sqrt[257 + 60*I])*x] - Log[34 + ((1 - 4*I) + Sqrt[257 + 60*I])*x]) In[4]:= FullSimplify[F[4] - F[0]] Out[4]= Pi - (1/2)*ArcTan[2752/825] In[5]:= N[%] Out[5]= 2.5018228707631676 In[6]:= Plot[F[x], {x, -4, 4}] ... This will propably work for "simple enough" (whatever that means) rational functions. Peter