Answer to Question #50231 in Complex Analysis for Asad

Question

Let f(z) be an analytic function in the annulus 0 <|z| < R for some positive real number R,Whose laurent series (in this annulus) is given by

f(z) = n from -∞ to ∞ ∑ { (-1)^n / (n^2)! ] } . Z ^ { 5n - n^2 -1}
A)) What Kind of Singularity is z=0 for f(z) ?
B)) Compute integral on Curve for [ z ^ 24 . f(z) dz] ,where C is a counterclockwise simple path lying in the annulus enclosing z=0
C)) Calculate Res (f) in z=0
D)) Evaluate Integral on Curve for [ sin Z .f(z) dz] , where C : |z| = (R/2) oriented counterclockwise
Note : please not by limit