The closed loop line integral-z-5z3-z2-8z-2dzevaluated counterclockwise- is

According to Cauchy's integral formula f(a)=12π j∫ _cf(z)(z a)dz (if f(z) is analytic within an on the closed curve C) Hence, ∫ _cf(z)(z a) where, f(z)=z^3+z^2+ ...

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The closed lo...

Question

The closed loop line integral
$\oint_{| z | = 5} \frac{z^{3} + z^{2} + 8}{z + 2} d z$
evaluated counterclockwise, is

- 4 j π

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+ 8 j π

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- 8 j π

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+ 4 j π

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Solution

The correct option is B $+ 8 j π$
According to Cauchy's integral formula
$f (a) = \frac{1}{2 π j} \int_{c} \frac{f (z)}{(z - a)} d z$
(if $f (z)$ is analytic within an on the closed curve $C$ )
Hence,
$\int_{c} \frac{f (z)}{(z - a)}$ {where, $f (z) = z^{3} + z^{2} + 8$ }
$= 2 π j f (- 2)$ { $∵ a = - 2$ }
$= 2 π j (- 8 + 4 + 8)$
$= + j 8 π$

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The closed loop line integral ∮|z|=5z3+z2+8z+2dz evaluated counterclockwise, is

The correct option is B +8jπAccording to Cauchy's integral formula f(a)=12πj∫cf(z)(z−a)dz (if f(z) is analytic within an on the closed curve C) Hence, ∫cf(z)(z−a) {where, f(z)=z3+z2+8} =2πjf(−2) {∵a=−2} =2πj(−8+4+8) =+j8π

The closed loop line integral
$\oint_{| z | = 5} \frac{z^{3} + z^{2} + 8}{z + 2} d z$
evaluated counterclockwise, is