A Taylor series expands a holomorphic function on a disc. A Laurent series expands a holomorphic function on an annulus, with both positive and negative powers.
Laurent expansion on an annulus (Tag T-4XPWL)
Let \(z_0\in\CC\), \(0\le r<R\le\infty\), and let \(f\) be holomorphic on the annulus \[ A\coloneqq\ts{z\in\CC\st r<\abs{z-z_0}<R}. \] Then \(f\) has a two-sided expansion \[ f(z)=\sum_{n\in\ZZ}c_n(z-z_0)^n \] converging locally uniformly on \(A\), with coefficients \[ c_n=\frac{1}{2\pi i}\int_{\abs{z-z_0}=\rho}\frac{f(z)}{(z-z_0)^{n+1}}\dz \] for any \(r<\rho<R\), independent of \(\rho\). The expansion is unique.
If \(r=0\), so that \(z_0\) is an isolated singularity of \(f\), then the principal part \(\sum_{n<0}c_n(z-z_0)^n\) has no nonzero terms if and only if \(z_0\) is removable, finitely many if and only if \(z_0\) is a pole, and infinitely many if and only if \(z_0\) is essential.
See [@Ahl79, Chapter 5, Section 1.3, p. 184].
Cauchy integral formula for Taylor coefficients and the Cauchy estimates (Tag C-7S2CO)
Let \(f\) be holomorphic on an open set containing the closed disc \(\overline{D_R(p)}\), and write \(f(z)=\sum_{k\ge0}c_k(z-p)^k\) near \(p\). Then for every \(k\ge0\), \[ c_k=\frac{f^{(k)}(p)}{k!}=\frac{1}{2\pi i}\int_{\abs{z-p}=R}\frac{f(z)}{(z-p)^{k+1}}\dz=\frac{1}{2\pi R^k}\int_0^{2\pi}f(p+Re^{i\theta})e^{-ik\theta}\dtheta. \] Consequently, with \(M_R\coloneqq\max_{\abs{z-p}=R}\abs{f(z)}\), \[ \abs{c_k}\le\frac{M_R}{R^k}, \] so \(\limsup_k\abs{c_k}^{1/k}\le1/R\), and the Taylor series of \(f\) at \(p\) converges on \(D_R(p)\), where it equals \(f\).
Differentiating the Cauchy integral formula \(f(w)=\frac{1}{2\pi i}\int_{\abs{z-p}=R}\frac{f(z)}{z-w}\dz\) under the integral sign \(k\) times at \(w=p\) gives the first two equalities. Parametrizing the circle by \(z=p+Re^{i\theta}\), so that \(\dz=iRe^{i\theta}\dtheta\), gives the third. Bounding the integrand of the third expression by \(M_R/R^k\) over an interval of length \(2\pi\) gives the estimate, and the root test gives the radius of convergence.
Dependence on the annulus
A function holomorphic on several disjoint annuli centered at \(z_0\) has a Laurent series about \(z_0\) on each of them, and the series can differ.
(One function, two expansions)
Let \(f(z) = {1 \over z-1}\), holomorphic on \(\CC\sm\ts{1}\), expanded about \(z=0\).
On \(\abs z < 1\), write it as a geometric series in \(z\): \[ {1\over z-1} = -{1\over 1-z} = -\sum_{k\geq 0} z^k .\]
On \(\abs z > 1\), factoring out \(z\) gives a geometric series in \(\inverseof{z}\): \[ {1\over z-1} = {1\over z}\cdot{1 \over 1 - \inverseof{z}} = \sum_{k \geq 1} z^{-k} .\]
Each series converges to \(f\) on its annulus.
Computing Laurent series
By uniqueness of the Laurent expansion on a given annulus, any convergent expansion obtained from known series is the Laurent series, and its coefficients equal the coefficient integrals.
Geometric series. Write the expression as \({1 \over 1 - u}\) with \(\abs u < 1\) on the given annulus, and expand. On \(\abs{z}<1\) one takes \(u\) a multiple of \(z\), and on \(\abs z>1\) a multiple of \(\inverseof{z}\).
Products and compositions of known expansions. For \(f = g\cdot h\) with \(g\) having a pole and \(h\) holomorphic, expand \(h\) as a Taylor series and multiply through by the finite principal part of \(g\).
Poles of known order. For \(f\) with a pole of order \(N\) at \(z_0\), \((z-z_0)^N f(z)\) has a removable singularity at \(z_0\); its Taylor series divided by \((z-z_0)^N\) is the Laurent series of \(f\) on a punctured disc.
Partial fractions. A rational function is a polynomial plus a sum of terms \(c/(z-a)^j\), and each term is expanded on the given annulus.
Residues and singularity type
On a punctured disc about \(z_0\), the coefficient \(c_{-1}\) is the residue of \(f\) at \(z_0\), which enters the residue theorem.
On a punctured disc about \(z_0\), the singularity is removable, a pole, or essential according as no, finitely many, or infinitely many coefficients \(c_k\) with \(k<0\) are nonzero (Classifying a singularity).
Exercises
Laurent series of $1/\sin z$ at $0$ (Tag FF-UQZNR)
For \(0<\abs{z}<\pi\), \[ {1\over \sin(z)} = \frac{1}{z}+\frac{1}{3 !} z+\frac{7}{360} z^{3}+O\qty{z^{5}}. \]
Laurent expansion of $\csch z$ at $0$ (Tag FF-5GLOZ)
The function \(\csch z\coloneqq\frac{1}{\sinh z}\) has a simple pole at \(0\), and for \(0<\abs{z}<\pi\) \[ \csch z=\frac1z-\frac{z}{6}+\frac{7z^{3}}{360}-\frac{31z^{5}}{15120}+\cdots. \]