B.2 Consider the following complex function TT CSC TZ T 1 f(2) w² - 22 w² - 2² sin 72¹ where w # Z may be complex. a) Fi

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B 2 Consider The Following Complex Function Tt Csc Tz T 1 F 2 W 22 W 2 Sin 72 Where W Z May Be Complex A Fi 1 (54.93 KiB) Viewed 37 times

please answer only a and b
B.2 Consider the following complex function TT CSC TZ T 1 f(2) w² - 22 w² - 2² sin 72¹ where w # Z may be complex. a) Find all its singularities, state their nature and compute the residues. [8] b) Consider the contour y displayed on Fig. 1, with vertices at ±(N+1) ± iN and N> w. Show, using Cauchy's residue theorem, that [3] N CSC T CSC TW I(N) = 1 2πi i fuos ² d² = (-1)" dz w2 -n² --N c) Assume now that N > 1. Show that, on the segment z = x+iN, -N- < x < N + 1, [4] 1 s sin Tz sinh(r) d) Similarly, show that, on the vertical segment z = : ± (N+ ¹) + iy, −N <y<N, [4] 1 ≤1. sin 72 cosh ty e) Using the bounds of sections d) and e), show now that lim I(N) = 0 and, therefore [6] N-00 00 TT CSC TW = w (-1)" w² -n² 12--00 -A* -N-1-N -A = -2 -1 N N+1