Question Number 136682 by snipers237 last updated on 24/Mar/21
$$\:{Im}\left(\int_{{C}^{+} \left(\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right)} \overset{−} {{z}dz}\:\right)=\:\frac{\pi}{\mathrm{2}}\:\: \\ $$
Answered by Olaf last updated on 24/Mar/21
$$\Omega\:=\:\mathrm{Im}\int_{\mathrm{C}^{+} \left(\mathrm{0},\frac{\mathrm{1}}{\mathrm{2}}\right)} \overset{−} {{z}dz} \\ $$$$\Omega\:=\:\mathrm{Im}\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{1}}{\mathrm{2}}{e}^{−{i}\theta} {d}\left(\frac{\mathrm{1}}{\mathrm{2}}{e}^{{i}\theta} \right) \\ $$$$\Omega\:=\:\frac{\mathrm{1}}{\mathrm{4}}\mathrm{Im}\int_{\mathrm{0}} ^{\mathrm{2}\pi} {id}\theta \\ $$$$\Omega\:=\:\frac{\mathrm{1}}{\mathrm{4}}\mathrm{Im}\left(\mathrm{2}\pi{i}\right)\:=\:\frac{\mathrm{2}\pi}{\mathrm{4}}\:=\:\frac{\pi}{\mathrm{2}} \\ $$