Laurent Series and Residues

Where a function fails to be holomorphic — at a singularity — it still has a series expansion, now including negative powers: the Laurent series. The coefficient of $1/(z-z_0)$ is the residue, and summing residues evaluates contour integrals by inspection. This is the computational heart of complex analysis and the workhorse of QFT loop integrals. This page builds on cauchy-integral.md; the applications are evaluating-integrals.md.

Laurent series

On an annulus $r<|z-z_0|<R$ (a disc with a possible singularity punched out at the centre), a holomorphic function has a unique Laurent expansion $$f(z)=\sum _{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n,a_n=\frac{1}{2\pi i}\oint \frac{f(z)}{(z-z_0{)}^{n+1}}dz.$$ The positive powers ($n\ge 0$) form the ordinary analytic part; the negative powers — the principal part — encode the singularity. When no negative powers appear the annulus fills in to a disc and Laurent reduces to Taylor.

Classifying singularities

An isolated singularity $z_0$ (holomorphic on a punctured disc) is classified by its principal part:

• Riemann's removable singularity theorem: boundedness near $z_0$ forces removability.
• A pole of order $m$ means $f(z)=g(z)/(z-z_0{)}^m$ with $g$ holomorphic and $g(z_0)\ne 0$; a simple pole is $m=1$.
• Casorati–Weierstrass / Picard: near an essential singularity $f$ comes arbitrarily close to every complex value — indeed, by the great Picard theorem (picard.md), it attains every value but at most one, infinitely often.

The residue

The residue of $f$ at $z_0$ is the coefficient of the $(z-z_0{)}^{-1}$ term: $${\mathrm{Res}}_{z_0}f=a_{-1}=\frac{1}{2\pi i}{\oint }_{\text{small loop}}fdz.$$ It is the only part of the Laurent series that survives integration (recall $\oint (z-z_0{)}^{n}dz=2\pi i{\delta }_{n,-1}$ from cauchy-integral.md). Practical formulas:

• Simple pole: ${\mathrm{Res}}_{z_0}f={\lim }_{z\to z_0}(z-z_0)f(z)$; for $f=p/q$ with a simple zero of $q$, this is $p(z_0)/q^{\prime }(z_0)$.
• Pole of order $m$: ${\mathrm{Res}}_{z_0}f=\frac{1}{(m-1)!}\underset{z\to z_0}{\lim }\frac{d^{m-1}}{d{z}^{m-1}}[(z-z_0{)}^{m}f(z)].$

The residue theorem

Summing the local computation over all enclosed singularities:

Residue theorem. If $f$ is holomorphic inside and on a positively oriented simple closed contour $\gamma$ except at isolated singularities $z_k$ inside, then $${\oint }_{\gamma }f(z)dz=2\pi i\sum _k{\mathrm{Res}}_{z_k}f.$$

A global integral is reduced to a finite sum of local data. Cauchy's theorem (integrand holomorphic ⇒ no residues ⇒ integral zero) and the integral formula (a single simple pole with residue $f(z_0)$) are both special cases. Everything in evaluating-integrals.md is an application.

The argument principle and Rouché

Applying the residue theorem to $f^{\prime }/f$ (whose residues count zeros and poles) gives:

Argument principle. $\frac{1}{2\pi i}{\oint }_{\gamma }>\frac{f^{\prime }(z)}{f(z)}dz=Z-P$, the number of zeros minus poles of $f$ inside $\gamma$ (with multiplicity).

Geometrically, this counts how many times $f(\gamma )$ winds around $0$. Its consequence Rouché's theorem ("if $|g|<|f|$ on $\gamma$ then $f$ and $f+g$ have the same number of zeros inside") localizes roots and gives a slick proof of the fundamental theorem of algebra. The winding-number viewpoint reappears physically as the counting of bound states / resonances via the phase of a scattering amplitude.

References

• Ahlfors, Complex Analysis, Ch. 5.
• Stein & Shakarchi, Complex Analysis, Ch. 3.
• Arfken, Weber & Harris, Mathematical Methods for Physicists, Ch. 11.