Can a complex function be grokked as a extending the y and x axis into complex planes rather than lines (requiring 4 dimensions), and the function is a line through this space?

Question

anonymous · Answer

I assume you mean outside the complex plane, right?

anonymous · Answer

I mean instead of  a point on the curve having co-ordinates (x,y), it has points (where y=a+ib and x=c+id) (a,b,c,d).

anonymous · Answer

Instead of a function z being seen as real values of x and y, you want to interpret a function z as complex valued instead? That's what I meant originally, beyond 2D...

anonymous · Answer

I know you can eg use the Bloch sphere as a way of viewing 2 complex numbers (4D) in quantum systems if that's any help......

anonymous · Answer

Bloch sphere projects a 'shadow' onto 3 dimensions? By what means?

anonymous · Answer

I will reply a bit later, OK?

experimentx · Answer

http://mathworld.wolfram.com/BranchCut.html http://en.wikipedia.org/wiki/Domain_coloring

anonymous · Answer

It's sort of awkward to explain properly.

Rotations in a 2D complex vector space contain a double representation of rotations in 3D real space (SU2 -> SO3 is 2:1).
To fit things into the sphere, we first ignore any global phase (the periodic aspect) and normalize the coefficients to 1.
I don't know if this is a sufficient explanation for you, if not, I can try to dig out the materials from my quantum computation course.....