English: Atan2 function. On horizontal axis 0X there is complex argument of point of unit circle. It is measured ( principal value ) from 0 to 2pi ( modulo 2 pi).

On vertical axis 0Y ther is output of function atan2 :

${\displaystyle y=atan2(z.re,z.im)}$

where ${\displaystyle z}$ is a point of unit circle :

${\displaystyle z=z.re+z.im\ast i}$

such

${\displaystyle z=\cos (t)+i\ast \sin (t)=e^{i\ast t}}$

so

${\displaystyle z.re=\cos (t)}$

${\displaystyle z.im=\sin (t)}$

${\displaystyle r=abs(z)=1}$

Atan2 function gives argument of complex number with principal value in range -pi to pi, , that is -π < atan2(y, x) ≤ π.

Such definition makes function single-valued, ^[1] but creates discontinuity at angle pi.

Maxima source code

It uses gnuplot through Maxima

 L(t):=exp(%i*t*2*%pi); /* unit circle parametrised with angle in turns */
 plot2d(
 [atan2(imagpart(L(x)),realpart(L(x)))],
 [x,0,1],
 [y,-2*%pi,2*%pi],
 [plot_format, gnuplot],
 [gnuplot_term,"png"],
 [gnuplot_out_file, "atan2.png"],
 [legend, "atan2"],
 [xlabel,"angle in radians"], 
 [ylabel,"angle in radians"],
 [gnuplot_preamble,"
   set key left top;
  set xtics ('pi/2' 0.25, 'pi' 0.5, '3pi/2' 0.75,'2pi' 1.0);
  set ytics ('-2pi' -6.283, '-pi' -3.1415, '-pi/2' -1.5708, '0' 0, 'pi/2' 1.5708, 'pi' 3.1415, '2pi' 6.283);
  set grid xtics ytics 
  "]
 );

After plotting in Maxima vertical line was manually removed ( using image editor) and red circles was added.