The value of the integral $\frac{1}{2 π j} \oint_{C} \frac{z^{2} + 1}{z^{2} - 1} d z$ where $z$ is a complex number and $C$ is a unit circle with center at $1 + 0 j$ in the complex plane is

The value of the contour integral
$\frac{1}{2 π j} \oint {(z + \frac{1}{z})}^{2} d z$
evaluated over the unit circle $| z | = 1$ is

Conisder the complex valued funciton $f (z) = 2 z^{3} + b | z |^{3}$ where $z$ is a complex variable. The value of $b$ for which the function $f (z)$ is analytic is

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The integral ∮unitcirclef(z)dz evaluated around the unit circle on the complex plane for f(z)=coszz is

The correct option is A 2πif(z)=coszz f(z) has a simple pole at z=0 ∮Cf(z)dz=2πi {Res.of f(z) at z=0} 2πi[limz→0zf(z)] =2πicos0=2πi

The integral $\oint_{u n i t c i r c l e} f (z) d z$ evaluated around the unit circle on the complex plane for $f (z) = \frac{c o s z}{z}$ is

The correct option is A $2 π i$
$f (z) = \frac{c o s z}{z}$
$f (z)$ has a simple pole at $z = 0$
$\oint_{C} f (z) d z = 2 π i$ {Res.of $f (z)$ at $z = 0$ }
$2 π i [{lim}_{z \to 0} z f (z)]$
$= 2 π i c o s 0 = 2 π i$