local rotations of complex functions at roots Suppose I have $f:\mathbb{C}\rightarrow\mathbb{C}$ with a root at $r$, and I want to make a $g(z)$ for which $r$ is also a root but $\lim_{z\rightarrow r} g(z)/f(z) =e^{i\theta}$ for some $\theta\in\mathbb{R}$
For example take $f(z)=z^{4}-1$, for which there is a phase portrait on the left below. (was generated with mpmath and matplotlib). The image on the right was made by applying the gimp's swirl option in the Iwarp distort to the root at $1$ -- consider it the phase portrait of $g(z)$: 


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*How does one construct $g(z)$ from $f(z)$ with prescribed rotations at each root?



 A: I wanted to post this as a comment, but it was too long. I'm not certain it does what is required, so I'll be happy to delete if it's not helpful.
Possibility #1: If $f(z)$ is analytic and you want $g(z)$ to be analytic:
If you want to rotate $f(z)$ at a finite number of roots $r_1, r_2, \cdots, r_n$, and you want to rotate by $a_k$ at root $r_k$, you might be able to use $g(z) = L(z) f(z)$, where $L(z)$ is a Lagrange polynomial.
$$L(z) = \sum \limits _{j = 1}^n a_k \frac{(z - r_1)(z-r_2)\cdots \widehat{(z-r_k)}\cdots(z-r_n)}{(r_k - r_1)(r_k-r_2)\cdots \widehat{(r_k-r_k)}\cdots(r_k-r_n)}$$
where $\widehat{(z-r_k)}$ and $\widehat{(r_k-r_k)}$ means those factors are not included in the product.
I think $L(r_k) = a_k$, and so at each root $r_k$
$$
  \lim \limits _{z \to r_k} \frac{g(z)}{f(z)}
   = \lim \limits _{z \to r_k} \frac{L(z) f(z)}{f(z)}
   = \lim \limits _{z \to r_k} L(z)
   = a_k
$$ 
Possibility #2: $g(z)$ does not need to be analytic: Let $B(r_k, \delta_k, z)$ be a bump function with compact support near a root $r_k$ with $B(r_k, \delta_k, r_k) = 1$ and $B(r_k, \delta_k, z) = 0$ when $|z - r_k| > \delta_k$.
Outside all $|z - r_k| < \delta_k$ disks let $g(z) = f(z)$.
Inside a $|z - r_k| < \delta_k$ disk let $g(z) = \Big[a_k B(r_k, \delta_k, z) + (1- B(r_k, \delta_k, z) ) \Big] f(z)$.
