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 (of course the conjugate of a root is always a root since
−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
mod z • exp(i • arg z).
ℂ→ℂ mappings mess with my head…and I like it.