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MathGroup Archive 1998

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Unexpected behaviour with Series/Integrate/Normal



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.



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