StudyBrixFree worksheets, study guides and courses

Mathematics · Advanced university

Residue at a second-order pole

Differentiate the analytic numerator once; its value p(a) is not the residue for a double pole.

Make a worksheet

Method and conditions

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 examples

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

Find the residue at z = 8 of f(z) = (1z³ + 3z)/(z − (8))². Round the final result to 4 decimal places if needed.

Differentiate the numerator: p′(z) = 3z² + 3. Evaluate at 8.

195

Try it

  1. Find the residue at z = 4 of f(z) = (9z³ + 5z)/(z − (4))². Round the final result to 4 decimal places if needed.

  2. Find the residue at z = 9 of f(z) = (12z³ + 7z)/(z − (9))². Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 437

    Differentiate the numerator: p′(z) = 27z² + 5. Evaluate at 4.

  2. 2923

    Differentiate the numerator: p′(z) = 36z² + 7. Evaluate at 9.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.

AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.