The residues of a complex function $${\text{X}}\left( {\text{z}} \right) = \frac{{1 - 2{\text{z}}}}{{{\text{z}}\left( {{\text{z}} - 1} \right)\left( {{\text{z}} - 2} \right)}}$$ at its poles are
The value of the integral $$\oint {\frac{{2{\text{z}} + 5}}{{\left( {{\text{z}} - \frac{1}{2}} \right)\left( {{{\text{z}}^2} - 4{\text{z}} + 5} \right)}}{\text{dz}}} $$ over the contour |z| = 1, taken in the anti-clockwise direction, would be
An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv(x, y), where $${\text{i}} = \sqrt { - 1} .$$ If u(x, y) = x2 - y2, then expression for v(x, y) in terms of x, y and a general constant c would be
The residue of the function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{{\left( {{\text{z}} + 2} \right)}^2}{{\left( {{\text{z}} - 2} \right)}^2}}}$$ at z = 2 is
The residue of $${\text{f}}\left( {\text{z}} \right) = \frac{{{{\text{z}}^3}}}{{{{\left( {{\text{z}} - 1} \right)}^4}\left( {{\text{z}} - 2} \right)\left( {{\text{z}} - 3} \right)}}$$ at z = 3 is
Given $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{{\text{z}} + 1}} - \frac{2}{{{\text{z}} + 3}}.$$ If C is a counter clockwise path in the z-plane such that |z + 1| = 1, the value of $$\frac{1}{{2\pi {\text{j}}}}\oint_{\text{C}} {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ is
