Consider the complex valued function $$f\left( z \right) = 2{z^3} + b{\left| z \right|^3}$$ where $$z$$ is a complex variable. The value of $$b$$ for which the function $$f(z)$$ is analytic is __________.

In the following integral, the contour $$C$$ encloses the points $${2\pi j}$$ and $$-{2\pi j}$$. The value of the integral $$ - {1 \over {2\pi }}\oint\limits_c {{{\sin z} \over {{{\left( {z - 2\pi j} \right)}^3}}}} dz$$ is ___________.

Let $$z=x+iy$$ be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction . Which one of the following statement is NOT TRUE?

If $$C$$ is a circle of radius $$r$$ with centre $${z_0}$$ in the complex $$z$$-plane and if $$'n'$$ is a non-zero integer, then $$\oint\limits_c {{{dz} \over {{{\left( {z - {z_0}} \right)}^{n + 1}}}}} $$ equals