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When we have a function like $f(x) = x^2-1$ and we expand it in a power series about some $x = x_0$, does Mathematica automatically assume that $x$ is real valued?

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    – bbgodfrey
    Oct 16 at 13:18
  • $\begingroup$ I do not believe so. However, if f[x] is not analytical x0, the result depends on the direction in the complex plain at which x0 is approached. See the documentation. $\endgroup$
    – bbgodfrey
    Oct 16 at 13:22
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    $\begingroup$ If you have a function like your example $f$, the power series will be the same polynomial whether we assume $x$ is real or complex. You might want to give an example in which it matters in the same way it matters in your actual problem. For instance, Abs[x]; try Series on it with and without the assumption that $x$ is real. $\endgroup$
    – Michael E2
    Oct 16 at 18:40
  • $\begingroup$ Thanks to both for your contribution. @Michael E2, I have tried with your example and indeed it has a difference in that case. My specific example turns out to be the one in my OP. I guessed that by expanding about x = 1, I'm implying to Mathematica that x is real. What do you think? $\endgroup$
    – dekees
    Oct 16 at 21:50
  • $\begingroup$ I don't think it implies x is real for Mathematica. What does it matter, though? The answer for x is real is the same as the answer for x is complex. You could add Assumptions -> Element[x, Reals], if you want to indicate that x may be assumed to be real. $\endgroup$
    – Michael E2
    Oct 17 at 2:05

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