Just playing with `z² / z² + 2z + 2`

on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.

As opposed to normal differentiability, ℂ-differentiability of a function implies:

- infinite descent into derivatives is possible (no chain of
`C¹ ⊂ C² ⊂ C³ ... Cω`

like usual)

nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)

Pretty interesting to just change things around and see how the parts work.

- The roots of the denominator are
`1+i`

and `1−i`

(of course the conjugate of a root is always a root since `i`

and `−i`

are indistinguishable)
- you can see how the denominator twists
- a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
- if you change the
`z^2/`

to a `z/`

or a `1/`

you can see that.
- then the Wikipedia picture shows the poles (infinities)

Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real"⊎"imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements `z`

are `mod z • exp(i • arg z)`

.

ℂ→ℂ mappings mess with my head…and I like it.