Complete Analysis About'RESIDUES AT SIMPLE POLE' With Solved Problem

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At a simple pole c, the residue of f is given by: If that limit does not exist, there is an essential singularity there. If it is 0 then it is either analytic there or there is a removable singularity. If it is equal to infinity then the order is higher.
Suppose f=P/Q is a rational function and suppose f has a simple pole at a. Then a formula for calculating the residue of f at a is Res(f(z),a)=limz→a(z−a)f(z)=limz→aP(z)Q(z)−Q(a)z−a=P(a)Q′(a)

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