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I wanted to know if we can have power series for functions like $x^{\alpha}$, where $\alpha \in \mathbb{R}$. One case we know is that $\alpha \in \mathbb{N} \cup \left\lbrace 0 \right\rbrace$, where $x^{\alpha}$ is already in the power series representation with all but one coefficients zero.

What about other values of $\alpha$? Can we still have a power series (NOT Taylor's series) for $\alpha \in \mathbb{R} \setminus \left( \mathbb{N} \cup \left\lbrace 0 \right\rbrace \right)$? If so, how do we find it?

Aniruddha Deshmukh
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  • That is my exact question. Because Taylor series expansion about $x = 0$ is not possible. – Aniruddha Deshmukh Jun 21 '19 at 06:26
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    Maybe don't expand about $x=0$ then? There are infinitely many points you can perform a valid Taylor series expansion about. – Peter Foreman Jun 21 '19 at 06:28
  • I want to have an expansion about $x = 0$ for some other application. If this is not possible, I will have to think of some completely different approach. – Aniruddha Deshmukh Jun 21 '19 at 06:30
  • You will not be able to perform an expansion about $x=0$ but you can perform an expansion about $x=\epsilon$ where $\epsilon\to0^+$. This expansion will then converge for $|x-\epsilon|\lt1$ – Peter Foreman Jun 21 '19 at 06:31

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You certainly can have Puiseux series, which are formal power series with the powers being rational instead of integers, like $$ \sqrt{z(2+z)}=\sqrt{2}\sum_{j=0}^\infty \binom{1/2}{j}2^{-j}z^{(2j+1)/2}. $$ We use them in, e.g., parametrising a curve near a branch point.

user10354138
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