Mathematics · Advanced university
Residue at a simple pole
The numerator is analytic near a and the pole is simple.
Method and conditions
The numerator is analytic near a and the pole is simple.
Res[p(z)/(z−a),a] = p(a)
Worked examples
Find the residue at z = 2 of f(z) = (2z² + 8)/(z − (2)). Round the final result to 4 decimal places if needed.
Multiply by z − (2) and take the limit z → 2. The remaining polynomial is evaluated at the pole.
16
Find the residue at z = -8 of f(z) = (3z² + 11)/(z − (-8)). Round the final result to 4 decimal places if needed.
Multiply by z − (-8) and take the limit z → -8. The remaining polynomial is evaluated at the pole.
203
Try it
Find the residue at z = -3 of f(z) = (6z² + 12)/(z − (-3)). Round the final result to 4 decimal places if needed.
Find the residue at z = 3 of f(z) = (3z² + 3)/(z − (3)). Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 66
Multiply by z − (-3) and take the limit z → -3. The remaining polynomial is evaluated at the pole.
- 30
Multiply by z − (3) and take the limit z → 3. The remaining polynomial is evaluated at the pole.
Further reading and review references
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