PYQs Based Test - 23 : Cauchy's Integral Theorem and Formula

Let $f(z)=\frac{1}{z^2+6z+9}$ defined in the complex plane. The integral ∮${}_c$ f(z)dz over the contour of a circle c with center at the origin and unit radius is ___.

The closed loop line integral _≤ft| z | = 5^ {z^3 + z^2 + 8}z + 2dz evaluated counter-clockwise, is

The value of the contour integral in the complex plane

$\oint \frac{z^3-2z+3}{z-2}dz$

along the contour |z| = 3, taken counterclockwise is

The values of the integral $\frac{1}{2\pi \mathrm{j}}\underset{\mathrm{c}}{\oint }\frac{{\mathrm{e}}^{\mathrm{z}}}{\mathrm{z}-2}$ 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

Let C represent the unit circle centered at origin in the complex plane, and complex variable, z = x + iy. The value of the contour integral ${\oint }_C^{}\frac{\cosh 3z}{2z}dz$ (where integration is taken counter clockwise) is