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Recently, I am searching for an operator $F$ that can filter out fractional powers of a Puiseux series $P(x)$, e.g. $$F[3x^{\frac12}+4x+10x^2-0.5x^\frac43+x^\frac73]=4x+10x^2$$

Assume an operator $F$ is linear and satisfies: $$F[kx^a]=k\cdot F[x^a]$$ $$F[x^a]= \begin{cases} 0, & \text{if $a$ is an integer} \\ \text{not 0}, & \text{otherwise} \end{cases}$$

and its inverse operator is linear and satisfies $$F^{-1}[F[kx^a]]=kx^a$$ for non-integer $a$ and define $F^{-1}[0]=0$.

Then $P(x)-F^{-1}[F[P(x)]]$ would filter out all fractional powers.

I thought $$F[x^a]=\oint_{|x|=1}x^adx=\frac{e^{2\pi ia}-1}{a+1}$$ would do the job but sadly, its inverse $$F^{-1}[2\pi i/w]=\frac{-W(-we^{-w})-w}{2\pi i}-1$$ is not linear.

Any idea?

Thanks in advance.

Szeto
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  • The keyword to search for is multisection. – Mariano Suárez-Álvarez Mar 31 '18 at 04:29
  • If you are considering formal Puiseux series, then your "integral representation" makes no sense — even if you think that they're over complex numbers, there are divergent series. On the other hand, if you are defining this operator on some (maybe topological) basis, then you can simply define it as 0 on Laurent monomials and identity otherwise. – xsnl Mar 31 '18 at 04:30
  • @MarianoSuárez-Álvarez Multisection could be a direction but I encounter difficulties when generalizing the multisection theorem from formal power series to Puiseux series. Could you kindly provide some advice? – Szeto Mar 31 '18 at 05:34

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