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Question Number 146334 by tabata last updated on 12/Jul/21

Commented by tabata last updated on 12/Jul/21

$$helpmesirquestiin1and3$$

Answered by Olaf_Thorendsen last updated on 12/Jul/21

$$\mathrm{Q}1.$$$${\mathrm{\Omega }}_1={\int }_{\mid z\mid =3}\frac{z^2}{(z-1)(z-2)}dz$$$${\mathrm{\Omega }}_1={\int }_{\mid z\mid =3}(1-\frac{1}{z-1}+\frac{4}{z-2})dz$$$${\mathrm{\Omega }}_1={\int }_{\mid z\mid =3}(1-\frac{1}{z-1}+\frac{4}{z-2})dz$$$${\mathrm{\Omega }}_1=0-2i\pi +4\times 2i\pi =6i\pi$$$$$$$$\mathrm{Q}3.$$$${\mathrm{\Omega }}_3={\int }_{\mid z-2\mid =3}\frac{e^z+\sin z}{z}dz$$$${\mathrm{Res}}_{0}f=e^0+\sin 0=1$$$${\mathrm{\Omega }}_3=2i\pi {\mathrm{Res}}_{0}f=2i\pi$$

Answered by mathmax by abdo last updated on 13/Jul/21

$$4){\int }_{\mid \mathrm{z}\mid =\frac{1}{2}}\frac{\mathrm{cosz}}{\mathrm{z}{(\mathrm{z}+\mathrm{i})}^2}\mathrm{dz}=2\mathrm{i}\pi \mathrm{Res}(\mathrm{f},\mathrm{o})=-2\mathrm{i}\pi \mathrm{because}$$$$\mathrm{Res}(\mathrm{f},0)=\frac{\cos 0}{{(0+\mathrm{i})}^2}=-1$$

Answered by mathmax by abdo last updated on 13/Jul/21

$$1){\int }_{\mathrm{C}}\frac{{\mathrm{z}}^2}{(\mathrm{z}-1)(\mathrm{z}-2)}\mathrm{dz}=2\mathrm{i}\pi \{\mathrm{Res}(\mathrm{f},1)+\mathrm{Res}(\mathrm{f},2)\}$$$$\mathrm{withf}\left(\mathrm{z}\right)=\frac{{\mathrm{z}}^2}{(\mathrm{z}-1)(\mathrm{z}-2)}\Rightarrow \mathrm{Res}(\mathrm{f},1)=\frac{1^2}{(1-2)}=-1$$$$\mathrm{Res}(\mathrm{f},2)=\frac{2^2}{(2-1)}=4\Rightarrow \mathrm{I}=2\mathrm{i}\pi \{-1+4\}=6\mathrm{i}\pi$$

Answered by mathmax by abdo last updated on 13/Jul/21

$$2)\mathrm{f}\left(\mathrm{z}\right)=\frac{\mathrm{z}}{({\mathrm{z}}^2-9)(\mathrm{z}+\mathrm{i})}\Rightarrow {\int }_{\mid \mathrm{z}\mid =2}\frac{\mathrm{zdz}}{(9-{\mathrm{z}}^2)(\mathrm{z}+\mathrm{i})}$$$$=-{\int }_{\mid \mathrm{z}\mid =2}\frac{\mathrm{zdz}}{(\mathrm{z}-3)(\mathrm{z}+3)(\mathrm{z}+\mathrm{i})}=-2\mathrm{i}\pi \mathrm{Res}(\mathrm{f},-\mathrm{i})$$$$=-2\mathrm{i}\pi .\frac{-\mathrm{i}}{(-1+9)}=\frac{-2\pi }{8}=-\frac{\pi }{4}$$

Commented by Mrsof last updated on 13/Jul/21

$$putsir:Res(f,-i)=\frac{-i}{10}$$

Answered by mathmax by abdo last updated on 13/Jul/21

$$3)\mathrm{I}={\int }_{\mid \mathrm{z}-2\mid =3}\mathrm{f}\left(\mathrm{z}\right)\mathrm{dz}=2\mathrm{i}\pi \mathrm{Res}(\mathrm{f},\mathrm{o})$$$$\mathrm{Res}(\mathrm{f},\mathrm{o})={\lim }_{\mathrm{z}\to 0}\mathrm{zf}\left(\mathrm{z}\right)={\lim }_{\mathrm{z}\to 0}{\mathrm{e}}^{\mathrm{z}}+\mathrm{sinz}=1\Rightarrow \mathrm{I}=2\mathrm{i}\pi$$

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