Complex Variables and Partial Differential Equations (3130005)

Question-3b-OR

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

Q3) (b)

Evaluate using Cauchy’s residue theorem $\int_{C} \frac{e^{2 z}}{{(z + 1)}^{3}} d z$ , Where, $C : 4 x^{2} + 9 y^{2} = 16$ .

Here,

f (z) = \frac{e^{2 z}}{{(z + 1)}^{3}}

&

C : 4 x^{2} + 9 y^{2} = 16

.

Res (f (z), z_{0}) = \frac{1}{(n - 1)!} \lim_{z \to z_{0}} [\frac{d^{n - 1}}{{dz}^{n - 1}} {(z - z_{0})}^{n} f (z)]

\Rightarrow Res (f (z), - 1) = \frac{1}{2!} \lim_{z \to - 1} [\frac{d^{2}}{{dz}^{2}} {(z + 1)}^{3} \frac{e^{2 z}}{{(z + 1)}^{3}}]

\Rightarrow Res (f (z), - 1) = \frac{1}{2} \lim_{z \to - 1} [4 e^{2 z}]

\Rightarrow Res (f (z), - 1) = 2 e^{- 2}

→ Then by Cauchy's residue theorem,

\int_{C} \frac{e^{2 z}}{{(z + 1)}^{3}} d z = 2 πi (Sum of residue) = 2 πi (2 e^{- 2})

\Rightarrow \int_{C} \frac{e^{2 z}}{{(z + 1)}^{3}} d z = 4 {πie}^{- 2}

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