2-quadrant and 4-quadrant inverse tangent
phi=atan(x)
phi=atan(y,x)
:x real or complex scalar, vector or matrix : :phi real or complex scalar, vector or matrix : :x, y real scalars, vectors or matrices of the same size : :phi real scalar, vector or matrix :
The first form computes the 2-quadrant inverse tangent, which is the inverse of tan(phi). For real x, phi is in the interval (-pi/2, pi/2). For complex x, atan has two singular, branching points +%i, -%i and the chosen branch cuts are the two imaginary half- straight lines [i, i*oo) and (-i*oo, -i].
The second form computes the 4-quadrant arctangent (atan2 in Fortran), this is, it returns the argument (angle) of the complex number x+i*y. The range of atan(y,x) is (-pi, pi].
For real arguments, both forms yield identical values if x>0.
In case of vector or matrix arguments, the evaluation is done element- wise, so that phi is a vector or matrix of the same size with phi(i,j)=atan(x(i,j)) or phi(i,j)=atan(y(i,j),x(i,j)).
// examples with the second form
x=[1,%i,-1,%i]
phasex=atan(`imag`_(x),`real`_(x))
atan(0,-1)
atan(-%eps,-1)
// branch cuts
atan(-%eps + 2*%i)
atan(+%eps + 2*%i)
atan(-%eps - 2*%i)
atan(+%eps - 2*%i)
// values at the branching points
`ieee`_(2)
atan(%i)
atan(-%i)