# atan

2-quadrant and 4-quadrant inverse tangent

## Calling Sequence

    phi = atan(x)
                phi = atan(y, x)

## Arguments

x

  a real or complex scalar, vector or matrix.

phi

  a real or complex scalar, vector or matrix.

x, y

  a real scalars, vectors or matrices of the same size.

phi

  a real scalar, vector or matrix.

## Description

The first form computes the 2-quadrant inverse tangent, which is the inverse of `tan(phi)`. For real `x`, `phi` is in the interval \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right). 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*\infty) and (-i*\infty, -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

    // examples with the second form
    x=[1,%i,-1,%i]
    phase_x=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)

## See Also

- tan
- unwrap
- ieee