Formula for argument

For \(z=x+iy\ne0\), choose the principal argument \(\theta\in(-\pi,\pi]\). A quotient \(y/x\) can be used only when \(x\ne0\), and its inverse tangent must be adjusted for the quadrant.

This is because it depends on which quadrant the complex number
\[ z = x+iy \]
lies.

Case 1: \(x>0\). Then \[\theta=\arctan(y/x).\] This covers both right-hand quadrants and the positive real axis.

Case 2: \(x<0\) and \(y\ge0\). Then \[\theta=\pi+\arctan(y/x),\] where \(\arctan(y/x)\in(-\pi/2,0]\). In particular, the negative real axis has principal argument \(\pi\).

Case 3: \(x<0\) and \(y<0\). Then \[\theta=-\pi+\arctan(y/x).\] On the imaginary axis, \(x=0,y>0\) gives \(\theta=\pi/2\), and \(x=0,y<0\) gives \(\theta=-\pi/2\). The argument of \(0\) is undefined.