Just playing with `z² / z² + 2z + 2`

$g(z)=frac{z^2}{z^2+2z+2}$

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.