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Assumptions - an issue. Does mathematica store it?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122527] Assumptions - an issue. Does mathematica store it?
  • From: Sumesh PT <sumesh at jncasr.ac.in>
  • Date: Mon, 31 Oct 2011 06:50:02 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

Hi, 
I have come across a peculiar situation in Mathematica where the same expression integrated twice with the same set of assumptions gives me two different answers (the answers are same on simplification). However I am wondering how can the software decided to do the integration in two different methods and does it have anything to do with the assumptions that once uses. An example code is given below. It might be a good idea to paste it in a note book so that you can know what I am talking about. Answers at1 and at2 in the code should be same ,but they look different. Can anybody explain it?

Thanks,
sumesh
In[35]:= ClearAll[x]

In[36]:= att = (25  (x^4) \[Rho] )/(4 (x^2 + \[Rho]^2)^(7/2)) - (
   75  (x^2) (\[Rho]^3) )/(16 (x^2 + \[Rho]^2)^(7/2)) - (
   75  (\[Rho]^5) )/(64  (x^2 + \[Rho]^2)^(7/2)) - (
   15  (x^2) \[Rho] )/(4  (x^2 + \[Rho]^2)^(5/2)) + (
   15  (\[Rho]^3) )/(8  (x^2 + \[Rho]^2)^(5/2)) - (
   75  (x^4) (zp^3) \[Rho] )/(
   8  (x^2 + \[Rho]^2)^(7/2) (zp^2 + \[Rho]^2)^(5/2)) + (
   75  (x^4) zp (\[Rho]^3) )/(
   16  (x^2 + \[Rho]^2)^(7/2) (zp^2 + \[Rho]^2)^(5/2)) + (
   75  (x^3) (zp^2) (\[Rho]^3) )/(
   4  (x^2 + \[Rho]^2)^(7/2) (zp^2 + \[Rho]^2)^(5/2)) + (
   225  (x^2) (zp^3) (\[Rho]^3) )/(
   16  (x^2 + \[Rho]^2)^(7/2) (zp^2 + \[Rho]^2)^(5/2)) + (
   225  (xzp^2) (\[Rho]^5) )/(
   32  (x^2 + \[Rho]^2)^(7/2) (zp^2 + \[Rho]^2)^(5/2));

In[37]:= at1 = 
 Integrate[att, {\[Rho], 0, \[Infinity]}, 
  Assumptions -> {x \[Element] Reals, x^2 > 0, zp^2 > 0}]

Out[37]= (1/(24 x^4 zp^4 (x^2 - 
   zp^2)^5))(-30 x^4 zp^4 (12 x^6 zp + 48 x^5 zp^2 + 80 x^3 zp^4 - 
     15 xzp^2 zp^4 - 3 x^4 (xzp^2 - 40 zp^3) - 
     30 x^2 zp^2 (xzp^2 - 2 zp^3)) + 
  Abs[x]^3 (30 zp^4 (10 x^7 + 60 x^6 zp + 90 x^5 zp^2 + 30 x^3 zp^4 - 
        3 xzp^2 zp^4 - 2 x zp^6 - 15 x^4 (xzp^2 - 8 zp^3) - 
        6 x^2 (5 xzp^2 zp^2 - 2 zp^5)) Abs[
       zp] + (x^2 - 
        zp^2)^5 (12 zp^4 Hypergeometric2F1[1, 2, -(3/2), x^2/
          zp^2] + (16 x^2 zp^2 - 21 zp^4) Hypergeometric2F1[2, 
          2, -(1/2), x^2/zp^2] + 
        32 x^4 Hypergeometric2F1[2, 3, 1/2, x^2/zp^2] - 
        168 x^2 zp^2 Hypergeometric2F1[2, 3, 1/2, x^2/zp^2] + 
        9 zp^4 Hypergeometric2F1[2, 3, 1/2, x^2/zp^2] + 
        504 x^4 Hypergeometric2F1[2, 4, 3/2, x^2/zp^2] - 
        108 x^2 zp^2 Hypergeometric2F1[2, 4, 3/2, x^2/zp^2] + 
        144 x^4 Hypergeometric2F1[2, 5, 5/2, x^2/zp^2])))

In[38]:= at2 = 
 Integrate[att, {\[Rho], 0, \[Infinity]}, 
  Assumptions -> {x \[Element] Reals && x^2 > 0 && zp^2 > 0}]

Out[38]= (1/(4 x^2 (x^2 - zp^2)^5))5 (-12 x^8 zp - 48 x^7 zp^2 - 
   80 x^5 zp^4 + 15 x^2 xzp^2 zp^4 + 3 x^6 (xzp^2 - 40 zp^3) + 
   30 x^4 zp^2 (xzp^2 - 2 zp^3) + (10 x^7 + 60 x^6 zp + 90 x^5 zp^2 + 
      30 x^3 zp^4 - 3 xzp^2 zp^4 - 2 x zp^6 - 
      15 x^4 (xzp^2 - 8 zp^3) + 6 x^2 zp^2 (-5 xzp^2 + 2 zp^3)) Abs[
     x] Abs[zp])



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