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MathGroup Archive 2007

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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


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