(mathstodon.xyz) Dan Piponi: "Sketch of proof based on paper…" - Mathstodon

ROAM_REFS: https://mathstodon.xyz/@dpiponi/114468239096746615

Sketch of proof based on paper linked in answer:

Define:

z_k+1 = √zⱼ
zⱼ = xⱼ+iyⱼ = rⱼexp(iθⱼ)
(Choosing √ and θⱼ so θ_j+1=θⱼ/2.)

A lemma I'll leave to you: lim (j→∞) 2ʲyⱼ = θⱼ

Define the process: f(x,r) = ((x+r)/2,√(r(x+r)/2)).

This is very similar to the computation of √(x+iy). In fact we have

f(xⱼ,rⱼ)/yⱼ = (x_j+1,r_j+1)/2y_j+1
(Treating (x,r) as a vector.)

We have A(x,r) = A(f(x,r)) and also A(ax,ar)=aA(x,r). So we get

A(x₀,r₀)/y₀ = A(x₁,r₁)/(2y₁) = lim (j→∞) A(xⱼ,rⱼ)/(2ʲyⱼ)

In the limit we get:

A(x₀,r₀)/y₀ = A(1,1)/θ₀.

Ie. A(x₀,r₀) = y₀/θ₀

I think it's really neat that each of x, y, r and θ appear once.

As an example, pick z=1+i. x=1, y=1, r=√2, θ=π/4. So A(1,√2)=4/π.

TL;DR The modified AGM is basically a complex square root operation which is being iterated until some numbers are close to 1 and then we can use linearity.