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.