Reference request: Where is this functional equation found? $$
g\left(\frac{x+y}{1+xy}\right) = g(x)g(y).
$$
One solution is
$$
g(x) = \frac{1+x}{1-x}.
$$
Another is
$$
g(x) = \sqrt\frac{1+x}{1-x}.
$$
Any other power of the first solution is also a solution, but for the moment I suspect the second one may be in some reasonable sense the natural one.
Is there a standard name of this functional equation?
Is there an extensive literature?
Does it occur naturally in some particular field?
Are there interesting results?  What are they?
 A: $\newcommand{\+}{^{\dagger}}%
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$\ds{\gg\pars{x + y \over 1 + xy} = \gg\pars{x}\gg\pars{y}:\ {\large ?}}$

\begin{align}
\gg'\pars{x + y \over 1 + xy}\,{1 - y^{2} \over \pars{1 + xy}^{2}} &= \gg'\pars{x}\gg\pars{y}
\\[3mm]
\gg'\pars{x + y \over 1 + xy}\,{1 - x^{2} \over \pars{1 + xy}^{2}} &= \gg\pars{x}\gg'\pars{y}
\end{align}
which leads to
$\ds{{1 - y^{2} \over 1 - x^{2}} = {\gg'\pars{x}\gg\pars{y} \over \gg\pars{x}\gg'\pars{y}}\quad\imp\quad
\pars{1 - x^{2}}\,{\gg'\pars{x} \over \gg\pars{x}}
=
\pars{1 - y^{2}}\,{\gg'\pars{y} \over \gg\pars{y}} = C}$. $C$ is a constant. Then,
\begin{align}
&\ln\pars{\verts{g}}
=\int{\dd\gg \over g}
= \half\,C\int{\dd x \over 1 - x^{2}} + \overbrace{D}^{\ds{\mbox{constant}}}
=
\half\,C\int\pars{{1 \over x - 1} - {1 \over 1 + x}}\,\dd x + D
\\[3mm]&=
\half\,C\ln\pars{\verts{x - 1 \over x + 1}} + D
\end{align}

$$
\ln\pars{\verts{\gg\pars{x}}} = \half\,C\ln\pars{\verts{x - 1 \over x + 1}} + D\,,
\qquad C, D: \mbox{constants}
$$
$$
\verts{\gg\pars{x}} = A\verts{x - 1 \over x + 1}^{C/2}\,,\qquad A:\mbox{constant}
$$


*

*$\gg\pars{0} = 0$ leads to the trivial solution
    $\gg\pars{x} = 0\,,\ \forall\ x$.

*$\gg\pars{0} = 1$ yields
    $\ds{\verts{\gg\pars{x}} = \verts{x - 1 \over x + 1}^{C/2}}$.

*What else ?. 

