PYQs Based Test - 25 : Residue Theorem and Solution Integral

The value of the contour integral $\frac{1}{2\pi j}\oint {\left(z+\frac{1}{z}\right)}^{2}dz$ evaluated over the unit circle |z| = 1 is _______.

If C is a circle |z| = 4 and $f\left(z\right)=\frac{z^2}{{\left(z^2-3z+2\right)}^2}$, then $\oint f\left(z\right)dz$ is

The value of the integral 1/2π i _C^ {z^2 + 1}{z^2 - 1}dz where z is a complex number and C is a unit circle with centre at 1 + 0j in the complex plane is __________ .

Let (-1 - j), (3 - j), (3 + j) and (-1 + j) be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the countour integral ${\oint }_C\frac{dz}{z^2(z-4)}$ is

Let $f\left(z\right)=\frac{1}{z+a},a>0$. The value of the integral ∮ f(z) dz over a circle C with center (-a, 0) and radius R > 0 evaluated in the anti-clock wise direction is