51. The value of the integral $$\oint {\frac{{{\text{z}} + 1}}{{{{\text{z}}^2} - 4}}{\text{dz}}} $$ in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is
52. The integral $$\oint {{\text{f}}\left( {\text{z}} \right){\text{dz}}} $$ evaluated around the unit circle on the complex plane for $${\text{f}}\left( {\text{z}} \right) = \frac{{\cos {\text{z}}}}{{\text{z}}}$$ is
53. The residues of a function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\left( {{\text{z}} - 4} \right){{\left( {{\text{z}} + 1} \right)}^3}}}$$ are
54. The value of expression $$\frac{{ - 5 + {\text{i}}10}}{{3 + 4{\text{i}}}}$$ is
55. The argument of the complex number $$\frac{{1 + {\text{i}}}}{{1 - {\text{i}}}},$$ where $${\text{i}} = \sqrt { - 1} ,$$ is
56. The nature of singularity of function $${\text{f}}\left( {\text{z}} \right) = \frac{1}{{\cos {\text{z}} - \sin {\text{z}}}}$$ at $${\text{z}} = \frac{\pi }{4}$$ is
57. A point z has been plotted in the complex plane, as shown in figure below.
The plot for point $$\frac{1}{{\text{z}}}$$ is
The plot for point $$\frac{1}{{\text{z}}}$$ is
58. The analytic function $${\text{f}}\left( {\text{z}} \right) = \frac{{{\text{z}} - 1}}{{{{\text{z}}^2} + 1}}$$ has singularities at
59. The values of the integral $$\frac{1}{{2\pi {\text{j}}}}\oint\limits_{\text{c}} {\frac{{{{\text{e}}^{\text{z}}}}}{{{\text{z}} - 2}}{\text{dz}}} $$ along a closed contour c in anti-clockwise direction for
i. the point z0 = 2 inside the contour c, and
ii. the point z0 = 2 outside the contour c, respectively, are
i. the point z0 = 2 inside the contour c, and
ii. the point z0 = 2 outside the contour c, respectively, are