(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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