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find-C-z-2-sin-z-2-dz-z-3pi-




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

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