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Mathematics · lesson 2 of 5

Poles, residues and contour integrals

The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Complex differentiability and Laurent series.

Residue

The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.

Simple pole

For p(z)/(z−a), with analytic p, the residue is p(a).

Contour theorem

A positively oriented simple closed contour integrates a meromorphic function to 2πi times enclosed residues, with no poles on the contour.

Orientation

Reversing contour direction reverses the sign of the integral.

Worked examples

Example 1

Find the residue of (z²+1)/(z−2) at z=2.

Evaluate the analytic numerator at 2: 4+1=5.

5

Example 2

Find the residue of z²/(z−1)² at z=1.

For the double pole, differentiate the numerator and evaluate: 2z at 1 gives 2.

2

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Find the residue of (z²+1)/(z−2) at z=2.

Show answer and explanation

5

Evaluate the analytic numerator at 2: 4+1=5.

2 Find the residue of z²/(z−1)² at z=1.

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2

For the double pole, differentiate the numerator and evaluate: 2z at 1 gives 2.

3 Which Laurent coefficient is the residue at a?

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The coefficient of (z−a)^(−1)

This is the definition of a residue.

4 Integrate 1/z counterclockwise around the unit circle. Enter the coefficient of 2πi.

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1

The only enclosed pole has residue 1.

5 Why does the standard residue formula need modification or a different setup when a pole lies on the contour?

Sample answer and marking guidance

The ordinary contour integral is not defined through a singularity in the usual way. One must specify an indentation, principal value or another limiting procedure before applying suitable results.

Credit the undefined ordinary path integral and the need for a declared treatment.

6 Integrate 3/z clockwise around the unit circle. Enter the coefficient of 2πi.

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-3

The residue is 3 and clockwise orientation changes the sign.

7 Find the residue of 1/[z(z−1)] at z=0 and explain the limit.

Sample answer and marking guidance

It is −1, because zf(z)=1/(z−1), which tends to −1.

Credit cancellation of the simple pole factor and evaluation.

8 If a function is holomorphic inside and on a closed contour in a simply connected region, what is its contour integral?

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0

Cauchy's integral theorem applies under these conditions.

Further reading and review references

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