Answer Happy

eiz 1 Consider first -dz, where y is a positively-oriented rectangular contour, with vertices ER and ER+iR, and an indentation at z = 0. a) Show that, on the limit R + oo, the contribution of the lateral and top segments to the integral vanish. [6] We will now consider instead I = sint eiwt dt, where w is a real constant. a [2] - b) Show how f(z) = {eilw+1)z is relevant to I. c) Using a semicircular contour over the upper half plane and an indentation at z = 0, show that, if |w| > 1, I = 0. [6] [2] d) What happens if w < -1? [7] e) Use appropriate indented semicircular contours to show that, if lw < 11, I = . [2] f) Finally, compute I if w = 1.