Practical Evaluation of Residues

The Laurent expansion is often not obvious in many practical applications, so additional procedures are needed. Often, the complex function appears as a ratio of polynomials 

if a simple pole exists at s = a, then obviously (s - a) must be a factor in gis), so we could express the denominator as g(s) = (s - a)G(s), provided G(s) contains no other singularities at s = a. Clearly, the Laurent expansion must exist, even though it may not be immediately apparent, and so we can always write a hypothetical representation of the type given in Eq. 9.75  

The procedure just outlined can be followed even if the pole cannot be factored out of g(s) in such a case, Eq. 9.78 would become 

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