Question Number 191 by 123456 last updated on 25/Jan/15
$$\mathrm{evaluate}\:\underset{\mathrm{C}} {\int}\frac{{z}^{\mathrm{2}} \mathrm{sin}\:{z}\:\mathrm{cos}\:{z}}{\left({z}^{\mathrm{2}} +\mathrm{1}\right)\left({z}^{\mathrm{2}} −\mathrm{1}\right)}{dz}\: \\ $$$$\mathrm{where}\:\mathrm{C}=\left\{{z}\in\mathbb{C}\mid\mid{z}\mid=\mathrm{2}\right\} \\ $$
Answered by nileshkulkarni last updated on 15/Dec/14
$$ \\ $$$$ \\ $$$${Apply}\:{Residue}\:{theorm}.\:{Poles}\:{of}\:{the} \\ $$$${fuction}\:{are}\:{z}=\:{i}.\:\:{z}=−{i}.\:\:{z}=\mathrm{1}.\:\:\:\:{z}=−\mathrm{1} \\ $$$$ \\ $$$${evaluating}\:{the}\:{fuction}\:{at}\:{these}\:{four} \\ $$$${poles}\:{and}\:{then}\:{adding}\:{it}\:{gives}\:\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$