If $$C$$ denotes the counter clockwise unit circle. The value of the contour integral $${1 \over {2\pi i}}\oint\limits_c {{\mathop{\rm Re}\nolimits} \left\{ z \right\}dz} $$ is __________.

Let $$f\left( z \right) = {{az + b} \over {cz + d}}.$$ If $$f\left( {{z_1}} \right) = f\left( {{z_2}} \right)$$ for all $${z_1} \ne {z_2}.\,\,a = 2,\,\,b = 4$$ and $$C=5,$$ then $$d$$ should be equal to

Given $$f\left( z \right) = {1 \over {z + 1}} - {2 \over {z + 3}}.$$ If $$C$$ is a counterclockwise path in the $$z$$-plane such that
$$\left| {z + 1} \right| = 1,$$ the value of $${1 \over {2\,\pi \,j}}\oint\limits_c {f\left( z \right)dz} $$ is

If $$x = \sqrt { - 1} ,\,\,$$ then the value of $${X^x}$$ is