Euler's Formula, but in the 3D Plane?!

https://docs.google.com/document/d/1QM-xt-EcnxO_9XKjAcafcI6wggcoxFiYgfYR6K33ykc/edit?tab=t.0

(This is David Chedester, aka. VergilChessDMC) 

I was just having fun with Euler's Formula, aka.  cos(a)+i sin(a), or commonly e^ia. When I set the Formula to a value greater than 1, I started getting some strange answers like for y=2, i got 2(πn - i tanh⁻¹(1/3)), n∈Z, where Z is the set of integers. So I thought to myself: What if I extended this to the 3D Plane? The google doc i've provided shows what happens. Let's just say it's not what anyone would've expected.

If one of y'all can come up with Proofs or Disproofs, let me know!

Note: I have to run the code via Replit for someone to see the visualization

nice