Question Number 147406 by tabata last updated on 20/Jul/21
$${find}\:\int_{{C}} \frac{{z}+\mathrm{2}}{{sin}\left(\frac{{z}}{\mathrm{2}}\right)}{dz}\:\:\:,\mid{z}\mid=\mathrm{3}\pi \\ $$
Answered by Olaf_Thorendsen last updated on 20/Jul/21
$$\Omega\:=\:\int_{\mid{z}\mid=\mathrm{3}\pi} \frac{{z}+\mathrm{2}}{\mathrm{sin}\left(\frac{{z}}{\mathrm{2}}\right)}\:{dz} \\ $$$$\Omega\:=\:\int_{\mid{z}\mid=\mathrm{3}\pi} \frac{{z}+\mathrm{2}}{{z}}×\frac{{z}}{\mathrm{sin}\left(\frac{{z}}{\mathrm{2}}\right)}\:{dz} \\ $$$$\Omega\:=\:\mathrm{2}{i}\pi\:\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{{z}\left({z}+\mathrm{2}\right)}{\mathrm{sin}\left(\frac{{z}}{\mathrm{2}}\right)}\right) \\ $$$$\Omega\:=\:\mathrm{2}{i}\pi\:\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{2}{z}+\mathrm{2}}{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\left(\frac{{z}}{\mathrm{2}}\right)}\right) \\ $$$$\Omega\:=\:\mathrm{2}{i}\pi\:×\mathrm{4}\:=\:\mathrm{8}{i}\pi \\ $$