Is there a name for this particular linear fractional transformation? Is there a conventional name for this function?
$$
\begin{align}
g(t) & = \frac{1+it}{1-it} \\[15pt]
& = \frac{1-t^2}{1+t^2} + i\frac{2t}{1+t^2}.
\end{align}
$$
This function comes up from time to time.
It enjoys these properties:


*

*(A restriction of) it is a homeomorphism from $\mathbb R\cup\{\infty\}$ to $\{z\in\mathbb C:|z|=1\}$ (where $\infty$ the infinity of complex analysis, so the former space is the one-point compactification of the real line).

*${}$
$$
\begin{align}
\text{If $x,y\in\mathbb R$ and $g(t)=x+iy$ then }g(-t) & = \phantom{-}x-iy, \\[10pt]
\text{and }g(1/t) & = -x+iy.
\end{align}
$$

*$\text{If $t=\tan\frac\theta2$ then }g(t) = e^{i\theta} = \cos\theta+i\sin\theta.\tag{1}$

*Via $(1)$ and the consequence that $d\theta=2\,dt/(1+t^2)$, it reduces certain differential equations involving rational functions of $\sin\theta$ and $\cos\theta$ to differential equations involving rational functions of $t$, and in particular reduces antidifferentiation of rational functions of $\sin\theta$ and $\cos\theta$ to antidifferentiation of rational functions of $t$.

*It is a bijection from $\mathbb Q\cup\{\infty\}$ to the set of rational points on the circle (thus showing that infinitely many of those exist, and consequently infinitely many primitive Pythagorean triples exist).

*As pointed out by "Did" in comments below, it is a conformal bijection between the Poincaré halfplane $\{z\in\mathbb C\mid \Im(z)>0\}$ and the open unit disk $\{z\in\mathbb C\mid |z|<1\}$ (and also between the complementary half-plane and the complement of the open unit disk).
Is there a conventional name for this function?

I'm not sure that this is what you have in mind, but here comes anyway.
If we look at the mapping $\phi:t\mapsto i g(t)$, we see that $\phi$ has $\pm1$ as fixed points. Furthermore $\phi:-i\mapsto\infty\mapsto i\mapsto 0\mapsto -i$. This paints a clear picture, as it means that $\phi$ is a 90 degree rotation of the Riemann sphere about the real axis. Your $g$ is thus a composition of two 90 degree rotations. This is another rotation, its axis can be found by finding the fixed points of $g$.
