Complex Variables

The function has singularities at and . We need to compute the residues of this function at both points.
Step 1: Residue at :
At , we have a pole of order 2. To find the residue, we use the formula for a second-order pole: Now, substitute : Differentiating with respect to : At , this becomes: Step 2: Residue at :
At , we have a pole of order 3. To compute the residue, we use the formula for the residue of a third-order pole: Substitute : Differentiating twice with respect to : At , this becomes: Thus, the residues are and .