A function $$f$$ of the complex variable $$z=x+iy,$$ is given as $$f(x,y)=u(x,y)+iv(x,y),$$
Where $$u(x,y)=2kxy$$ and $$v(x,y)$$ $$ = {x^2} - {y^2}.$$
The value of $$k,$$ for which the function is analytic, is __________.

The value of the integral $$\int\limits_{ - \infty }^\infty {{{\sin x} \over {{x^2} + 2x + 2}}} dx$$
evaluated using contour integration and the residue theorem is

If $$z$$ is a complex variable, the value of $$\int\limits_5^{3i} {{{dz} \over z}} $$ is

An analytic function of a complex variable $$z = x + iy$$ is expressed as
$$f\left( z \right) = u\left( {x + y} \right) + iv\left( {x,y} \right),$$ where $$i = \sqrt { - 1} .$$ If $$u(x, y)=$$ $${x^3} - {y^2}$$
then expression for $$v(x,y)$$ in terms of $$x,y$$ and a general constant $$c$$ would be