Answer Happy

please answer only d, e and f, thank you
B.1 Let a>0, a € R, n ≥ 2, n E N. Consider first the complex function f(x) = 1 2n + an a) Find all its singularities, state their nature and compute the residues. [5] b) Let y be the positively oriented sector contour given by the segment 0 < x < R, the arc z = Red, 0 < 0 < and the segment z = :re, with o = now evaluate 2. Assuming R > a, n 1 $ fost -dz. 2n + an [2] c) Considering the limit R→ ∞, use the previous result to show that dz π 1 S xn + an nan-1 sin ( n [10] d) Consider now the complex function g given by g(z) = = log z 2n + an The contour y of part c) needs an indentation. State why. [2] e) Make g(z) single-valued by specifying a branch cut, and compute the residue of g(z) for all singularities inside y. [2] f) Compute the contribution of the indentation, assuming it is circular and of radius P→ 0 [4] lim log z 2n + an -dz. P+0 ce