StudyBrixFree worksheets, study guides and courses

Mathematics · Advanced university

Poles, residues and contour integrals

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

Make a worksheet

Prerequisites

Complex differentiability and Laurent series.

Study notes

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

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

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

5

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

  1. Which Laurent coefficient is the residue at a?

    1. The constant coefficient
    2. Always the highest positive power
    3. The coefficient of (z−a)^(−1)
    4. The coefficient of (z−a)^(−2)

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

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

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

Answers and marking guidance

Answers and marking notes
  1. The coefficient of (z−a)^(−1)

    This is the definition of a residue.

  2. 1

    The only enclosed pole has residue 1.

  3. 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.

  4. -3

    The residue is 3 and clockwise orientation changes the sign.

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.