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Question Number 129688 by Engr_Jidda last updated on 17/Jan/21

$$complexanalysis$$$${\oint }_C\frac{{\varrho }^{2z}+{sinz}^2}{(z-2)(z-3)}dzC:\mid Z\mid =5$$

Answered by mathmax by abdo last updated on 17/Jan/21

$$\phi \left(\mathrm{z}\right)=\frac{{\mathrm{e}}^{2\mathrm{z}}+\sin \left({\mathrm{z}}^2\right)}{(\mathrm{z}-2)(\mathrm{z}-3)}\mathrm{the}\mathrm{poles}\mathrm{of}\phi \mathrm{are}2\mathrm{and}3$$$${\int }_{\mid \mathrm{z}\mid =5}\phi \left(\mathrm{z}\right)\mathrm{dz}=2\mathrm{i}\pi \{\mathrm{Res}(\phi ,2)+\mathrm{Res}(\phi ,3)\}$$$$=2\mathrm{i}\pi \{\frac{{\mathrm{e}}^4+\sin \left(4\right)}{-1}+\frac{{\mathrm{e}}^6+\sin 9}{1}\}=2\mathrm{i}\pi \{{\mathrm{e}}^6-{\mathrm{e}}^4+\sin 9-\sin 4\}$$

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