Mathematics · Advanced university
Poles, residues and contour integrals · revision notes
The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.
Key ideas and conditions
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.
Explain without looking
- Find the residue of 1/[z(z−1)] at z=0 and explain the limit.
- If a function is holomorphic inside and on a closed contour in a simply connected region, what is its contour integral?
Further reading and review references
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