(1) The Laurent series is:
1/z^2 + 1/(2z) - 5/6 - (11z)/24 + (101z^2)/120 + (331z^3)/720 + O(z^4)
(2) The principal part is non-zero, but only has finitely many terms. Therefore, the singularity of f at z = 0 is a pole order 2.
(3) The principal part is (-1 + e^i)/(2*(z-1)). This singularity is removable.
Hope this helps!
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