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The Higher Order Formula for Evaluating Residues

Ifis a function with a pole of order two at a pointthe Laurent series aboutcan be written in the form

If we want to find the termwe could multiply bythen lettend tobut the first term would mean this is undefined. Instead, multiplygiving

If we differentiate this equation, we obtain

Taking the limit as z tends togives us the residue at

In general, ifhas a pole of orderthen multiplybyto give

Differentiate thistimes to give

Now lendtend toto give

Example: Find the residue of the function

has a pole of order 4 atso