Complex Variables and Partial Differential Equations (3130005)

Question-3b

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

Q3) (b)

For, $f (z) = \frac{1}{{(z - 1)}^{2} (z - 3)}$ , find the residue at $z = 1$ .

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

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}{1!} \lim_{z \to 1} [\frac{d^{1}}{{dz}^{1}} {(z - 1)}^{2} \frac{1}{{(z - 1)}^{2} (z - 3)}]

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

\Rightarrow Res (f (z), 1) = - \frac{1}{4}

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