I am interested in finding a complex function that has a certain special case attached to it. Let $z_1$ and $z_2$ be complex numbers that lie on the surface of the unit circle. That is, $$ z_1 = \cos{(\theta_1)} +i \sin{(\theta_1)},\quad z_2 =\cos{(\theta_2)}+i \sin{(\theta_2)}. $$ The values for $\theta_1$ and $\theta_2$ are any arbitrary value within the interval of $(-\pi, \pi).$

Let $*$ denote some binary operation. What I am looking for is a non-constant function $w = f(z_1) * f(z_2)$ that will either output or equal $-1$ or $1$ on the real number line. There is a unique restraint that is necessary for the problem that I am solving, and can be expressed as such. The function needs to be expressed in terms of one-variable functions that are then added, multiplied, or subtracted from each other: $$ f(z_1)*(z_2) = f(z_1)\pm f(z_2) \space \space \text{or} \space \space f(z_1) \times f(z_2) \space \space $$ or any combination of addition, subtraction, and multiplication of separate terms.

Each function must include exclusively either $z_1$ or $z_2$. I have attempted functions such as: $$ \frac{1}{\ln(\frac{z_1}{z_2})^2+1} \qquad\text{or}\qquad 1-\frac{2}{e^{\ln(\frac{z_1}{z_2})^2}}.$$

While these functions do grant a final result that approaches either $-1$ or $1$, I have found no way to factor or break them up into separate terms of $f(z_1)$ and $f(z_2$). For instance, the root that I had for $ \frac{1}{\ln(\frac{z_1}{z_2})^2+1}$ was equal to $$\frac{1}{((\ln(z_1)-\ln(z_2))^2+i)((\ln(z_1)-\ln(z_2))^2-i)}$$ and I could not find a way to split it into separate transformations on $z_1$ and $z_2$. If you know of any function that satisfies the criteria, then I would be extremely grateful. Even if you don't know the exact solution that might solve this, recommending any class of functions, research reference, or salient reading source would still be appreciated.

  • $\begingroup$ what does "$f(z_1,z_2)$" mean??? $\endgroup$ Aug 5 at 20:47
  • $\begingroup$ It denotes a function that takes two complex numbers $z_1$ and $z_2$ as inputs. $\endgroup$
    – Nate
    Aug 5 at 22:45
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    $\begingroup$ Presumably the aim is to find a non-constant function of two variables that has the stated behavior and can be expressed in terms of one-variable functions by arithmetic operations and composition...? (If that's right, clarifying the question statement would help. On a related note, do the one-variable functions need to be holomorphic?) $\endgroup$ Aug 6 at 20:21
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    $\begingroup$ Doesn't $f(z_1, z_2) = z_1z_2$ satisfy your requirements? Since $z_1, z_2$ are on the unit circle, they have to be either $1$ or $-1$ to be on the real line. $\endgroup$ Aug 6 at 20:32
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    $\begingroup$ @Nate. Yes, $z_1, z_2$ can be any angle in general, but the requirement you stated is "$f(z_1, z_2) = \pm 1$ on the real number line". The only points on the unit circle that are also on the real number line are $1$ and $-1$. So your condition as stated amounts to $f(\pm 1, \pm 1) = \pm 1$. If that is not what you meant, then please clarify the requirement. $\endgroup$ Aug 7 at 5:46


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