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$\varphi =\arg(z)={\begin{cases}\arctan \left({\dfrac {y}{x}}\right)&{\text{if }}x>0\\\arctan \left({\dfrac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0\\\arctan \left({\dfrac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0\\{\dfrac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0\\-{\dfrac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0\\{\text{indeterminate }}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}$
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