Let \( C \) be a clockwise oriented closed curve in the complex plane defined by \( |z| = 1 \).
Further, let \( f(z) = jz \) be a complex function, where \( j = \sqrt{-1} \).
Then,
\[
\oint_C f(z)\, dz = \underline{2cm} \quad \text{(round off to the nearest integer)}.
\]
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Solution and Explanation
Given \( f(z) = jz \), this function is analytic (entire) everywhere in the complex plane.
Since \( f(z) \) is analytic inside and on the closed contour \( C \), by Cauchy's theorem: \[ \oint_C f(z) \, dz = 0 \] Direction of traversal (clockwise or counter-clockwise) does not matter if the function is analytic over the region enclosed. \[ \boxed{0} \]
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Two units, rated at 100 MW and 150 MW, are enabled for economic load dispatch. When the overall incremental cost is 10,000 Rs./MWh, the units are dispatched to 50 MW and 80 MW respectively. At an overall incremental cost of 10,600 Rs./MWh, the power output of the units are 80 MW and 92 MW, respectively. The total plant MW-output (without overloading any unit) at an overall incremental cost of 11,800 Rs./MWh is ___________ (round off to the nearest integer).
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