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I asked wolfram alpha to compute the antiderivative of the function $x^x$. It gave me some really large confusing polynomial-esque thing called a puiseux series. However, from what I can gather on the internet, the puiseux series is used to approximate points on a function. So what is the puiseux series and what does it have to do with integration? Also could you please explain it so that it would be understandable by someone who's highest mathematics taken was calculus BC.

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A Puiseux series is formal Laurent power series in $T^{\frac1n}$ for some $n$ (the $n$ may vary with the series). The set of Puiseux series is a field, denoted $k{\ll} T{\gg}$, and it is the algebraic closure of the field of formal power series $k[[T]]$.

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    $\begingroup$ ... except that in this case you have a generalized Puiseux series with logarithmic terms rather than fractional powers. Essentially that's because $$x^x = \exp(x \log(x)) = \sum_{n=0}^\infty \dfrac{x^n \log^n(x)}{n!}$$ $\endgroup$ – Robert Israel May 6 '16 at 1:27
  • $\begingroup$ @Robert Israel: I forgot to check the actual case. I gave the definition. used in, say, algebraic geometry. What you mention is simply an asymptotic expansion of $x^x$ along a scale of comparison (not sure of the English terminology), which is for me a more general concept than that of a Puiseux series. $\endgroup$ – Bernard May 6 '16 at 8:57

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