Saturday, October 31, 2020

Happy Halloween from Nested Tori!

Spooky greetings. A quick post with more complex analysis for you. Though maybe not as visually interesting as fractals, but getting back to some basics and/or nuance regarding complex functions. For one thing, I just never had a chance to really fit in a good study of Riemann surfaces (interest in them was partially stirred on by all that recent cubic hackery). Also, a lot of stuff involving branch cuts and branch points are something quickly gone over in standard complex analysis and never really explored in depth, other than to say "we take the principal branch of the logarithm/root/power" function to be... and you don't get into the nuance with the details of differences between different branches of functions. This example here is exploring the mapping $z \mapsto \sqrt{z^2 + 1}$. This is already interesting from the Riemann surface point of view, because neither the function nor its inverse is definable in its most interesting form by mapping (subsets of) the complex plane alone. Formally, to work on $\mathbb{C}$, we have to define things by taking a continuous choice of square root on $\mathbb{C}$ minus some ray from $0$ to $\infty$. The choice here is between a square root and its opposite (which is also a square root of the same complex number!) is not natural in the complex plane (unlike in the real numbers where you can always take the positive square root of a positive number, which is in some sense, a defining characteristic of real numbers, cementing why it is the unique complete ordered field), and being forced to choose forces a discontinuity somewhere. Riemann surface theory is, in some sense, about never having to make the choice. To quote Sir Roger Penrose (congrats on that Nobel Prize!): 

In particular, the domain of the logarithm function would be ‘cut’ in some arbitrary way, by a line out from the origin to infinity. To my way of thinking, this was a brutal mutilation of a sublime mathematical structure. (The Road to Reality, p. 136)

A couple of references (e.g. Penrose's student Tristan Needham, in his otherwise fantastic Visual Complex Analysis) suggest the most common branch cut for this is function is two horizontal lines to the left of $\mathsf{i}$ and $-\mathsf{i}$. I wrangled with this a bit, but I believe a subtlety is left out. To study $\sqrt{z^2 + 1}$ as truly the composition of a square root function, and the mapping $z^2 + 1$, your branch cuts will have to be the corresponding inverse image of the branch cut of the square root, under $z^2 + 1$. Using the negative real axis, this is the ray above $\mathsf{i}$ and the ray below $-\mathsf{i}$. Two rays to the left of $\pm \mathsf{i}$ cannot be realized as such an inverse image, for any ray to infinity (the image under $z^2 + 1$ is a whole parabola to infinity). Of course, the essential definition for complex analysis is that you define the function in this set by taking paths from an origin point and analytically continuing, and entrusting your result to algebraic topology. But this is not very effective, computationally. This is basically saying that for some selections of branch cuts, the function composition $\sqrt{z^2 + 1}$ is a lie: you cannot get it by assembling square root, squaring, and plus one. What turns out to work, in terms of assembling functions rather than continuing along curves, is not $\sqrt{z^2 + 1}$, but $\sqrt{z+\mathsf{i}} \sqrt{z-\mathsf{i}}$, that is, if you take the branch of square root, along the negative axis, subtract and add $\mathsf{i}$, take the square root of each, and then multiply, you do get something that reproduces what you theoretically get by paths. One might wonder what is the difference between $\sqrt{z^2 + 1} = \sqrt{(z+\mathsf{i})(z-\mathsf{i})}$ and $\sqrt{z+\mathsf{i}}\sqrt{z- \mathsf{i}}$; these are not the same because $\sqrt{ab} \neq \sqrt{a}\sqrt{b}$ in the complex plane unless you regard it as a multivalued set equality.

The best rendering of this function is, I've discovered, is taking the branch $-\mathsf{i}$ to $\mathsf{i}$ along the imaginary axis—something that doesn't even go out to infinity. To his credit, Needham does mention that connecting the two branch points in a finite part of the plane is possible, and it just has to be done in a manner that disables you from being able to complete a turn around the two branch points $\pm \mathsf{i}$. But there's no explicit computation that develops a visceral understanding of this fact (and so I made this post).

The title picture is the image of a large disk (parametrized by the usual polar coordinates) under the mapping $\sqrt{z-\mathsf{i}} \sqrt{z+ \mathsf{i}}$. The cut (deleted out of the disk) is taken on the interval $-\mathsf{i}$ to $\mathsf{i}$ ($y=-1$ to $y=1$ in the picture). Computationally this means the square roots were taken with the angles in the interval $-\pi/2$ to $3\pi/2$, which required me to define the square root using some hackery of the arctan function. I leave you with the conformal version of this (where the concentric ring radii increase exponentially rather than linearly; the linear increase looks better for looking like a cobweb, but this version looks better in preserving the proportions of the grid rectangles formed by the intersecting lines).

Or, take a look at it in Desmos:

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