Are there holomorphic coordinate transformations which stretch imaginary axis but compress real axis? Suppose we have a meromorphic function $g$, which has singularities at $\pm ia$ and $b$, where $0<a<b$. Clearly, convergence of its Taylor series around $0$ is limited by the singularities at $\pm ia$.
Is there such a holomorphic coordinate transformation $f$ (which maps $0$ to $0$) that radius of convergence of Taylor series around $0$ of $g\circ f$ would be limited by the singularity of $g$ at $b$ rather than the singularity at $\pm ia$?
In other words, are there such holomorphic functions $f$, which "compress" real axis but "stretch" imaginary axis when used as coordinate transformation? If yes, what are some examples?
 A: Consider a concrete function $f$ defined as
$$f(x)=\frac1{(x-3)^2}+\frac1{x^2+2}.$$
It's obvious that it has two simple poles at $x=\pm2i$ and a double pole at $x=3$. As the imaginary poles are closer to origin than the real one, Taylor series have radius of convergence limited by them, so $r=2$. See the following plot: blue line is $f$, orange is partial sum of Taylor series with $570$ terms:

Now let's introduce a function $m$ to transform the complex plane in such a way that the simple poles are moved farther than the double one:
$$m(x)=\frac1{8-2x}-\frac18.$$
See how this function transforms the function $t(z)=\sin(4\pi|z|)^{16}$:

As you can see, the second circle gets "expanded" so that its top is considerably farther from origin than its right hand part. So the points which were at the same distance before transformation are now moved.
Now after we expand $f\circ m$ to Taylor series, we'll get the following result (here the green curve is the partial sum of $f\circ m$):

Note though that it converges much more slowly than if we didn't have the poles we wanted to avoid and didn't change the variables. This is because the complex plane is effectively expanded at left and compressed at right, and the Taylor polynomials thus can approximate more closely the function at the left-hand side of the initial complex plane than that at the right-hand side.
