Evaluate the integral I integralAB z2 dz from zA 0 to zB

Evaluate the integral I = integral_A^B z^2 dz from = z_A = 0 to z_B = 1 + i, (a) along the contour Gamma_1 : y = x^2, (b) along y-axis from 0 to i, then along the horizontal line from i to 1 + i, as Gamma_2 shown in Fig. 2.7. Fig. 2.7. Two contour Gamma_1 and Gamma_2 from A (z_A = 0) to B (z_B = 1 + i), Gamma_1 : along the curve y = x^2, F_2 : first along y-axis to C (z_C = i), then along a horizontal line to B

Solution

Ans-

The answer to (a) is 10 by Proposition 18.1. The answer to (b) is e 2 , as one can restart the Poisson process at any event. The answer to (c) is P(S10 > 20) = P(N(20) < 10), so can either write the integral P(S10 > 20) = Z 20 et(t) 9 9! dt, or use P(N(20) < 10) = X 9 j=0 e 20 (20) j 9! . To answer (d), we condition on the number of events in [3, 4]: X 2 k=0 P(2 events in [1, 4] and 3 events in [3, 5] | k events in [3, 4]) · P(k events in [3, 4]) = X 2 k=0 P(2 k events in [1, 3] and 3 k events in [4, 5]) · P(k events in [3, 4]) = X 2 k=0 e 2 (2) 2k (2 k)! · e 3k (3 k)! · e k k! = e 4 1 3 5 + 4 + 1 2 3 .