$$f\left( z \right) = u\left( {x,y} \right) + i\,\,\,\,v\left( {x,y} \right)$$ is an analytic function of complex variable $$z=x+iy$$ , where $$i = \sqrt { - 1} $$ If $$u(x,y)=2xy,$$ then $$v(x,y)$$ may be expressed as

Given two complex numbers $${z_1} = 5 + \left( {5\sqrt 3 } \right)i$$ and $${z_2} = {2 \over {\sqrt 3 }} + 2i,$$ the argument of $${{{z_1}} \over {{z_2}}}$$ in degrees $$i$$

An analytic function of a complex variable $$z=x+iy,$$ where $$i = \sqrt { - 1} $$ is expressed as
$$f\left( z \right) = u\left( {x,y} \right) + i\,v\left( {x,y} \right).\,$$ If $$u(x,y)=2xy,$$
then $$v(x,y)$$ must be

The argument of the complex number $${{1 + i} \over {1 - i}},$$ where $$i = \sqrt { - 1} ,$$ is