Htykuut

If I have a complex function $f$ whose zeroes make curves in the complex plane, is there a way of getting the possibly-infinite length of all the curves that $f$ traces out? In particular, is there a nice way of defining this curve-length-measuring operator that's friendly to symbolic manipulation?

Consider the complex function $mathbb{C} to mathbb{C}$ like $f(z) = overline{z}cdot z - 1$ .

Its zeroes trace out a unit circle with length $2pi$ .

Let $w : (mathbb{C} to mathbb{C}) to mathbb{R}$ get the length of the zeroes.

$w$ seems to have some nice properties:

$f$ and $g$ are functions of $x$.

scaling or conjugating the result doesn't affect the length of the kernel.

$$ w(alpha cdot f) = w(f) $$
$$ w(overline{f}) = w(f) $$

Multiplying two functions takes the union of their zeroes.

$$ w(f cdot g) le w(f) + w(g) $$

translating the input doesn't change the length of the zeroes.

$$ w(f(z + x)) = w(f(x)) $$

I'm trying to figure out a nice closed form for the total length of all the curves traced out by the zeroes of a complex function $f$ provided that the function has derivatives in the neighborhood of all its zeroes and isn't constantly zero on any open set in $mathbb{C}$ .

Here is my first attempt at producing a definition for $w$ . There are probably a bunch of problems with it. I'm curious what the actual way of nailing down something like $w$ would be, but I think it does a good job of illustrating the intent.

The intuition is that, by imagining smaller and smaller squares around our zeroes, we can approximate the length of the zero curve inside the square by getting its "horizontal length" and "vertical length". Because we insisted on differentiability at the zeroes, as the squares get smaller our linear approximation gets better and better.

In order to actually complete the definition I'm using a somewhat bizarre zero-detecting function $Z_f(x, a)$ that traces a line in the complex plane from $x$ to $x+a$ and returns $1$ if there's a zero on the line and $0$ otherwise, so:

$$ Z_f(x, a) stackrel{df}{=} [exists ; 0 le t le 1 mathop{.} f(x + ta) = 0]$$

Here's a way of defining $w(f)$, the length of the curves traced by the zeroes of $f$, using a limit and $Z$ .

The total length is the horizontal and vertical length combined.

$$ w(f) stackrel{df}{=} sqrt{small{L} cdot overline{small{L}}} $$

$$ small{L} stackrel{df}{=}
sum_{a=(-infty)}^{infty}
sum_{b=(-infty)}^{infty} I + Jiota $$

$I$ is the horizontal length, computed by running over the width of a grid square and determining the "amount" of vertical lines that cross through a zero.

$$ I stackrel{df}{=} int_{a}^{a+m} Z_{f*}(s+biota, miota) mathop{ds} $$

$J$ is similar to $I$, but for the vertical length rather than horizontal.

$$ J stackrel{df}{=} int_{a}^{a+m} Z_{f*}(a+siota, m) mathop{ds} $$

I don't know if this is really necessary, but $f*$ is always a non-negative real, which seems like a friendlier function to hand off to our $Z$ operator.

$$ f^{*}(z) stackrel{df}{=} f(z) cdot overline{f(z)} $$

So, what's the preferred way to get the length of the curves traced by the zeroes of a complex function?