# When exactly is the principal AGM equal to the optimal AGM?

Definitions

Following the terminology of SageMath, let the principal arithmetic-geometric mean, $$\operatorname{AGP}$$, of $$(a,b)\in\mathbb{C}^2$$ for $$a\ne 0$$, $$b\ne 0$$, $$a\ne\pm b$$ be defined as follows: $$a_0=a,\, b_0=b,\\ a_{n+1}=\frac{a_n+b_n}{2},\,b_{n+1}=\sqrt{a_n b_n},\\ \operatorname{AGP}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty} b_n,$$ where at each $$b$$-iteration, $$\sqrt{}$$ is chosen such that $$\operatorname{Re}b_n\ge 0$$ for $$n\gt 0$$. This corresponds to always taking the so-called "principal square root" (which is usually denoted just by $$\sqrt{}$$).

Similarly, define the optimal arithmetic-geometric mean, $$\operatorname{AGO}$$, of $$(a,b)\in\mathbb{C}^2$$ for $$a\ne 0$$, $$b\ne 0$$, $$a\ne\pm b$$ as follows: $$a_0=a,\, b_0=b,\\ a_{n+1}=\frac{a_n+b_n}{2},\,b_{n+1}=\pm\sqrt{a_n b_n},\\ \operatorname{AGO}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty} b_n,$$ where at each $$b$$-iteration, a $$+$$ or a $$-$$ sign is chosen such that $$|a_n-b_n|\le |a_n+b_n|$$ for $$n\gt 0$$. If $$|a_n-b_n|=|a_n+b_n|$$, there is an ambiguity, so in case that happens, it is also required that $$\operatorname{Im}(b_n/a_n)\gt 0$$. Note that the condition which solves the ambiguity if $$|a_n-b_n|=|a_n+b_n|$$ is either not implemented in the algorithm used in SageMath, or it is implemented but not documented on their official website.

Following the terminology of Cox, a pair of sequences $$\{a_n\}_{n=0}^\infty$$ and $$\{b_n\}_{n=0}^\infty$$ is good if $$b_{n+1}$$ is the right choice for $$\pm \sqrt{a_n b_n}$$ for all but finitely many $$n\ge 0$$. The right choice is exactly what $$\operatorname{AGO}$$ chooses at each $$b$$-iteration.

Now define the multivalued arithmetic-geometric mean as $$\operatorname{AGM}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n$$ whenever $$\{a_n\}_{n=0}^\infty$$ and $$\{b_n\}_{n=0}^\infty$$ are good sequences. It is a complex multi-function of two variables which has a countable number of values.

Let $$\mathbb{H}=\{\tau\in\mathbb{C}:\,\operatorname{Im}\tau\gt 0\}$$ and $$\theta_3 (\tau ,0)=1+2\sum_{n=1}^\infty \mathrm{q}^{n^2}=p (\tau),$$ $$\theta_4 (\tau ,0)=1+2\sum_{n=1}^\infty (-1)^n\mathrm{q}^{n^2}=q (\tau)$$ where $$\mathrm{q}=e^{\pi i\tau}$$ is the elliptic nome.

An interesting theorem

Cox uses modular functions to prove the following theorem:

Let $$|a|\ge |b|$$. Then all possible values of $$\operatorname{AGM}(a,b)$$ are given by $$\frac{1}{\operatorname{AGM}(a,b)}=\frac{d}{\operatorname{AGO}(a,b)}+i\frac{c}{\operatorname{AGO}(a-b,a+b)}$$ where $$d$$ and $$c$$ are any relatively prime integers such that $$d\equiv 1\pmod 4$$ and $$c\equiv 0\pmod 4$$.

To see the connection between modular functions and $$\operatorname{AGM}$$, one can observe that $$p^2(2\tau)$$ is the arithmetic mean of $$p^2(\tau)$$ and $$q^2(\tau)$$, and that $$q(2\tau)$$ is the geometric mean of $$p(\tau)$$ and $$q(\tau)$$. Cox's paper explores the connection in greater detail.

Question

There is a non-empty set in $$\mathbb{C}^2$$ where $$\operatorname{AGP}(a,b)=\operatorname{AGO}(a,b)$$. What is it exactly?

Some examples: $$\color{green}{\operatorname{AGP}(2+3i,1+i)\approx 1.466+1.871i,}\,\color{green}{\operatorname{AGO}(2+3i,1+i)\approx 1.466+1.871i}$$ $$\color{red}{\operatorname{AGP}(-0.95-0.65i,0.683+0.747i)\approx 0.338-0.0135i,}\,\color{red}{\operatorname{AGO}(-0.95-0.65i,0.683+0.747i)\approx -0.371+0.319i}$$ An example with a $$1$$ is the following: $$\operatorname{AGP}(1,-2-i)\ne\operatorname{AGO}(1,-2-i)$$.

Motivation

$$\operatorname{AGO}$$ has the nice relation to the values of $$\operatorname{AGM}$$ in the complex plane, which the $$\operatorname{AGP}$$ seems to be lacking. The "interesting theorem" produces a multitude of values and it seems pretty hard to distill $$\operatorname{AGP}$$ from them.

Note also that the homogenity relation $$\lambda f(a,b)=f(\lambda a,\lambda b)$$ holds for the real arithmetic-geometric mean, but not necessarily for complex $$\operatorname{AGP}$$. For example, let $$a=-0.95-0.65i$$ and $$b=0.683+0.747i$$. Then, curiously, $$a\operatorname{AGP}\left(1,\frac{b}{a}\right)=\operatorname{AGO}(a,b)\ne\operatorname{AGP}(a,b).$$ This is caused by the fact that $$w\sqrt{z}=\sqrt{w^2z}$$ does not hold for some $$w,z\in\mathbb{C}$$.

There is also a remarkable connection between $$\operatorname{AGO}$$ and the hypergeometric function which is taken from Fungrim: Let $$(a,b)\in\mathbb{C}^2$$ and $$a/b\notin (-\infty ,0]$$. Then $$\operatorname{AGO}(a,b)=\frac{a+b}{2\, _2\mathrm{F}_1 \left(\frac{1}{2},\frac{1}{2},1,\left(\frac{a-b}{a+b}\right)^2\right)}.$$

Complex plots

Using the color function

I made the following plots: The horizontal axis represents $$\operatorname{Re}z$$ and the vertical axis represents $$\operatorname{Im}z$$.

This question was also asked on MSE.

• @DanieleTampieri: Did you read the question to the very end? :-) Mar 13, 2021 at 13:13
• @MateuszKwaśnicki Yep! It seems that I'm missing something 😁. Apologies to everyone! Mar 13, 2021 at 13:23
• Comment about the ambiguity: when I have implemented the AGO, I have resolved the ambiguity using fungrim.org/entry/a2b0f9. It's an arbitrary choice, but it's at least a documented arbitrary choice. Mar 14, 2021 at 8:23
• @FredrikJohansson Thank you for sharing, I couldn't find that on Sage Reference. By the way, I know this is not the best place to discuss this, but I think you have an error here: fungrim.org/entry/93831d. In the numerator, there should be $2^{n+1}$ instead of $2^n$.
– Wane
Mar 14, 2021 at 11:47
• @Wane Fixed, thanks! Mar 15, 2021 at 19:14