where a, b € R, and b > 0 and −1 < a < 2. za = Consider the complex function f(z) = 23 +6³ a) Justify why there is no ar

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Where A B R And B 0 And 1 A 2 Za Consider The Complex Function F Z 23 6 A Justify Why There Is No Ar 1 (124.27 KiB) Viewed 31 times

where a, b € R, and b > 0 and −1 < a < 2. za = Consider the complex function f(z) = 23 +6³ a) Justify why there is no arbitrarily small disc centered at the origin D(0, e) where f(z) analytic. b) Find all the branch points and isolated singularities of f(z), including at z = ∞, stating the order of the singularities. c) Adopt the branch cut - +0.01 < arg z < π + 0.01, and compute the residue of f(z) at all the singularities. d) Let y be the positively oriented, indented sector contour given by the segment p < x < R, the arc z = Reo, 0 <0 < 2, the segment z = xe²″i/³, ,p< x < R, and the small arc z = peio, 0 < 0 < ²5. Assuming R >1 and p < 1, evaluate 3 $₁3 23+ bada. -dz. Show that, in the limit R→ ∞ and p→ 0, the contribution of both circular arcs to the contour integral vanishes. f) Use the results of sections d) and e) to show that xa ㅠ 1 T₁ 2³²+ bide = d. x3 36²-ª sin [(1 + a)]