Consider Ir= eiz/z dz where S R is t he upper semi-circle as described above. If z = Reit , compute dz/dt . Use this to find dz in terms of dt. Show that |IR| 2 e-R sin(t) dt. Use the estimate sin(t) 2t/pi when 0 pi/2 to conclude that |IR| pi/R(1-e-R). Conclude that IR = 0.
Show transcribed image textConsider Ir= eiz/z dz where S R is t he upper semi-circle as described above. If z = Reit , compute dz/dt . Use this to find dz in terms of dt. Show that |IR| 2 e-R sin(t) dt. Use the estimate sin(t) 2t/pi when 0 pi/2 to conclude that |IR| pi/R(1-e-R). Conclude that IR = 0.