*To*: mathgroup@smc.vnet.net*Subject*: [mg11942] Unexpected behaviour with Series/Integrate/Normal*From*: Adrian Cable <megalith@jimc.demon.co.uk>*Date*: Thu, 9 Apr 1998 00:33:51 -0400*Organization*: MeGALiTH Software Ltd

Hi there, I've found an aspect of Mathematica 3.0.1's behaviour that I don't quite understand. It's possible that there's a bug somewhere, although I think it's most likely that there's a very reasonable explanation for this behaviour which I just can't see. Consider: In[31]:= Series[x^x,{x,0,10}] Out[31]= 1 + Log[x] x + (1/2)(Log[x]^2)(x^2) + (1/6)(Log[x]^3)(x^3) + ... + O[x]^11 In[32]:= Normal[%] Out[32]= 1 + Log[x] x + (1/2)(Log[x]^2)(x^2) + (1/6)(Log[x]^3)(x^3) + ... In[33]:= Integrate[%,x] Out[33]= x - (x^2)/4 + (x^3)/27 - (x^4)/256 + (x^5)/3125 + ... + P[x] Log[x] + Q[x](Log[x]^2) + ... (P[x] and Q[x] are polynomials of x with some rather large coefficients.) Now, this is fine. This is pretty much the result I'd expect. However, now consider: In[33]:= Series[x^x,{x,0,10}] Out[33]= 1 + Log[x] x + (1/2)(Log[x]^2)(x^2) + (1/6)(Log[x]^3)(x^3) + ... + O[x]^11 In[35]:= Integrate[%,x] Out[35]= x + (1/2)Log[x](x^2) + (1/6)(Log[x]^2)(x^3) + ... In fact, it turns out that for any integer k, (Series[x^x,{x,0,k}]-1)/(Log[x]) = Integrate[Series[x^x,{x,0,k}]]. Now, this result doesn't make sense, because it would imply that Integrate[x^x,x] = (x^x - 1)/Log[x], whereas in fact Integrate[x^x,x] cannot be expressed in closed form. What have I (or Mathematica, for that matter) done wrong here to get this obviously incorrect result? If anyone needs clarification here, I can supply a Mathematica notebook that shows this clearly. Thanks, cheers, Adrian Cable.