I am trying to design a group of complex functions $\rho_\alpha$ that have a type of symmetry that might look nice if it exists. This is what "symmetry" I want to try.
$$\alpha=a+bi\space\space,\space\space\bar\alpha=a-bi$$ $$\rho_\alpha(z)=\rho_\alpha(\alpha z+\bar\alpha)\tag{1}$$ $$\rho_\alpha'(0)=\frac{\alpha}{|\alpha|}\tag{2}$$
My questions are does this group of complex functions exist? How can I plot these complex functions using Python, JavaScript, or another programing language? Do they have nice properties?
I don't understand a lot about complex analysis only bits and pieces. I'm in high school just going into $11$th grade. I love math and I love art. I'm not looking for an exact formula for these functions just how I can compute them, but if there is an exact formula that would be good.
As Saad in the comments noticed, your condition can be simplified to $\rho(z)=\rho(az)$. This makes the definition near 0 tricky, so let’s ignore discontinuities there. Letting $f(z)=\rho(e^z)$, we get that $f$ is holomorphic and periodic in $2\pi i$ and $ln(a)$, so it’s an elliptic function. Elliptic functions are a nice class of well understood holomorphic functions with neat properties.