Answer to Question #235597 in Complex Analysis for Fozia Sayda

Question

find the residue of f(z) =1/(z²+1)² at z=i

Answer 1

$f(z) = \frac{1}{{{{\left( {{z^2} + 1} \right)}^2}}} = \frac{1}{{{{\left( {z + i} \right)}^2}{{\left( {z - i} \right)}^2}}}$

z = i is the zero of the order 2 denominator, so z = i is the second order pole.

Then

$\mathop {res}\limits_{z = i} f(z) = \mathop {\lim }\limits_{z \to i} {\left( {\frac{1}{{{{\left( {z + i} \right)}^2}{{\left( {z - i} \right)}^2}}} \cdot {{\left( {z - i} \right)}^2}} \right)^\prime } = \mathop {\lim }\limits_{z \to i} {\left( {\frac{1}{{{{\left( {z + i} \right)}^2}}}} \right)^\prime } = \mathop {\lim }\limits_{z \to i} \frac{{ - 2}}{{{{(z + i)}^3}}} = \frac{{ - 2}}{{{{(2i)}^3}}} = \frac{{ - 2}}{{ - 8i}} = \frac{1}{{4i}} = - \frac{i}{4}$

Answer: $\mathop {res}\limits_{z = i} f(z) = - \frac{i}{4}$