Mathematics · Advanced university
Residue at a second-order pole · formula reference
Differentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
Formula
Differentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
Res[p(z)/(z−a)²,a] = p′(a)
Worked example
Find the residue at z = -3 of f(z) = (10z³ + 11z)/(z − (-3))². Round the final result to 4 decimal places if needed.
Differentiate the numerator: p′(z) = 30z² + 11. Evaluate at -3.
281
Prerequisites and checking
Algebra; calculus or linear algebra as appropriate to the topic. Keep units and rounding consistent. State the model conditions before applying the formula.
Further reading and review references
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