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

The integral $$\oint {f(z)dz} $$ evaluated around the unit circle on the complex plane for $$f(z) = {{\cos z} \over z}$$ is

$${i^i}$$, where $$i\, = \,\sqrt { - 1} $$ is given by