Complex Variables and Partial Differential Equations (3130005)

Question-3a-OR

BE | Semester-3 Summer-2020 | 27-10-2020

Q3) (a)

Here,

f (z) = \frac{z + 4}{z^{2} + 2 z + 5}

and

C : |z + 1| = 1

Now,

z^{2} + 2 z + 5 = 0

\Rightarrow a = 1, b = 2, c = 5

\Rightarrow z = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a} = \frac{- 2 \pm \sqrt{4 - 20}}{2} = \frac{- 2 \pm \sqrt{- 16}}{2} = \frac{- 2 \pm 4 i}{2} = - 1 \pm 2 i

\Rightarrow f (z)

is not analytic at

z = - 1 + 2 i, - 1 - 2 i

Now,

C : |z + 1| = 1

|z + 1| = |(- 1 + 2 i) + 1| = |2 i| = 2 > 1 .

|z + 1| = |(- 1 - 2 i) + 1| = |2 i| = 2 > 1 .

\Rightarrow z = - 1 \pm 2 i are outside of C .

\Rightarrow f (z)

is analytic in

C

Then by Cauchy's Integral Theorem,

\int_{C} \frac{z + 4}{z^{2} + 2 z + 5} d z = 0