User:Svjo/sandbox

Derivation of atan2(y, x)

For the derivation of the principal value of the argument

The derivation of the principal value of the argument:

${\text{If}}\ (x,\,y)=(r\cos \theta ,r\sin \theta )\ {\text{then}}\ \tan {\cfrac {\theta }{2}}={\cfrac {y}{r+x}}$

It follows that

$\arg(x,y)=\theta =2{\theta \over 2}=2\arctan {y \over {\sqrt {x^{2}+y^{2}}}+x}\ .$

Note that √x² + y² + x ≠ 0 in the domain in question. Computation gives

$\nabla \arg(x,y)=\left({-y \over x^{2}+y^{2}},\ {x \over x^{2}+y^{2}}\right)\ .$