Laurent Series and Isolated Singular Points

Laurent Series

Motivation: What to Do with Singular Points?

In the neighborhood of a point where a function is not defined or not holomorphic, the Taylor series is not applicable—it requires holomorphicity in a disk. The Laurent series expands this by including negative powers: $f(z) = \dots + c_{-2}/(z-z_0)^2 + c_{-1}/(z-z_0) + c_0 + c_1(z-z_0) + \dots$ The principal part (with negative powers) describes the "type" of singularity.

Laurent Series

In an annulus $r < |z - z_0| < R$, a function $f$ that is holomorphic in the annulus is expanded as:

$ f(z) = \sum_{n=-\infty}^{\infty} c_n (z - z_0)^n $

where the coefficients are: $c_n = \frac{1}{2\pi i} \oint_{|\zeta - z_0| = \rho} \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} d\zeta$ ($r < \rho < R$).

Regular part: $\sum_{n=0}^\infty c_n (z - z_0)^n$—converges in the disk $|z - z_0| < R$. Principal part: $\sum_{n=1}^\infty c_{-n}/(z - z_0)^n$—converges outside the disk $|z - z_0| > r$.

Classification of Singular Points

Removable singularity: Principal part = 0 (all $c_{-n} = 0$). $f$ is bounded near $z_0$. It is possible to extend, defining $f(z_0) = c_0$ and continue to a holomorphic function. Example: $f(z) = \sin(z)/z$ at $z_0 = 0$. $\sin(z)/z = 1 - z^2/6 + z^4/120 - \dots$ (all exponents $\geq 0$).

Pole of order $m$: $c_{-m} \neq 0$, $c_{-n} = 0$ for $n > m$. Then $|f(z)| \rightarrow \infty$ as $z \rightarrow z_0$. $f(z) = (z-z_0)^{-m}\cdot g(z)$, where $g$ is holomorphic and $g(z_0) \neq 0$. Example: $1/(z-z_0)^m$—pole of order $m$.

Essential singularity: Infinitely many nonzero $c_{-n}$. Behavior is chaotic: no limit as $z \rightarrow z_0$, the image of $f$ in any neighborhood of $z_0$ is dense in $\mathbb{C}$ (Casorati–Weierstrass theorem). Example: $e^{1/z}$ at $z = 0$. Series: $e^{1/z} = 1 + 1/z + 1/(2! z^2) + \dots$

Meromorphic function: Holomorphic except at isolated poles. Rational $P(z)/Q(z)$—meromorphic on $\mathbb{C}$.

Casorati–Weierstrass Theorem

Near an essential singularity $z_0$, the image of $f$ is dense in $\mathbb{C}$: for any $w \in \mathbb{C}$ and any neighborhood $U \ni z_0$, there exists $z \in U$ such that $|f(z) - w| < \varepsilon$. Moreover (the stronger Picard’s theorem): $f$ takes every value (except, possibly, one) infinitely many times.

Numerical Example

Problem: Find the Laurent series of $e^{1/z}$ in the neighborhood of $z = 0$ and classify the singular point.

Step 1. Substitute $w = 1/z$: $e^w = \sum_{n=0}^\infty w^n/n!$

Step 2. Substitute $w = 1/z$: $ e^{1/z} = \sum_{n=0}^\infty \frac{1}{n! \cdot z^n} = 1 + \frac{1}{z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \dots $

Step 3. Principal part: $\sum_{n=1}^\infty 1/(n! \cdot z^n)$—infinitely many nonzero terms with negative exponents. Therefore, $z = 0$ is an essential singularity.

Step 4. Verifying the Casorati theorem: find $z \rightarrow 0$ with $e^{1/z} \rightarrow -1$. Take $1/z = i\pi + \ln(2) + 2\pi i k \rightarrow z = 1/( \ln 2 + i(\pi + 2\pi k) )$. As $k \rightarrow \infty$: $z \rightarrow 0$ and $e^{1/z} = e^{\ln 2} \cdot e^{i\pi} = 2 \cdot (-1) = -2$. Changing $\ln 2 \rightarrow 0$: $e^{1/z} \rightarrow -1$. Indeed, for any $w \neq 0$, one can find $z \rightarrow 0$ with $e^{1/z} \rightarrow w$.

Real-world Application

Quantum mechanics: essential singularities arise in particle scattering at zero energy ($s$-wave scattering). The classification of singularities determines the type of scattering (resonant or not).

Additional Aspects

The classification of singular points determines the behavior of integrals and applicability of residue theory. A pole of order $m$ is characterized by the fact that $(z-z_0)^m \cdot f(z)$ has a removable singularity; at such points the residue is computed by the formula $\operatorname{res} = \frac{1}{(m-1)!} \lim \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]$. Essential singularities (Sokhotski–Weierstrass theorem) yield a dense image in any neighborhood and are encountered in number theory and quantum mechanics problems. In practice, Laurent series are the basis for the partial fraction method when integrating rational functions, and for the contour method of computing inverse $z$-transforms in digital signal processing.

The outcome of this entire section becomes a unified algorithm: for any contour integral with a rational or meromorphic function, one should find all poles inside the contour, compute the residues, and sum them, multiplying by $2\pi i$. This recipe is transferred almost unchanged to integrals along the real axis by closing with an arc and checking Jordan's estimates; its universality explains why residue theory remains the principal computational tool of classical analysis.

Connection with Other Areas of Mathematics

Laurent series naturally arise in the study of linear differential equations in the complex domain. Frobenius’ approach to singular points uses expansions of solutions in a series with possibly negative powers, and the classification of singular points of the equation (ordinary, regular, irregular) according to Fuchs relies on the behavior of the coefficients in the Laurent expansion. In monodromy theory, analytic continuation of the solution around a singular point is described by a matrix, whose elements are expressed through the coefficients of the principal part.

The algebraic side appears in the theory of functions on Riemann surfaces. Meromorphic functions on a compact Riemann surface form a field, and their behavior in the neighborhood of points is defined by a finite number of terms in the Laurent series: this underlies divisors and the Riemann–Roch theorem. In algebraic geometry, rational differentials are described by the principal parts at each point, and the residue sum theorem is formulated as a linear relation among the coefficients of the principal parts.

In topology and theory of special values of Mellin–Barnes integrals, Laurent expansions are used in proofs of functional equations for zeta functions and $L$-functions; Riemann’s work on the zeta function essentially relies on the analysis of principal parts at points $1$ and $0$. In probabilistic problems associated with generators of Markov processes and generating functions, Laurent expansions allow one to study the asymptotics of the distribution tails through the analysis of singularities of the generating function. In numerical analysis, the Padé method and generalized Chebyshev expansions implement rational approximations whose structure is closely related to the location of poles, i.e., to the principal parts of Laurent series.

Historical Note and Development of the Idea

Pierre Laurent himself published in 1843 in Comptes Rendus a note describing expansion of functions near a point using negative powers. However, this construction received its systematic place later, in the works of Cauchy and Weierstrass, when the Cauchy integral began to be used for determining the coefficients of the series. The classical form of the expansion via an integral formula was fixed in courses by Serret and Goursat in the second half of the 19th century. By the end of the 19th century, thanks to works by Casorati, Weierstrass, and Picard, the idea of Laurent expansion became a key to understanding complex singularities and to formulating Picard’s theorem about the values of meromorphic functions. The works of Fuchs and Poincaré on linear differential equations introduced systematic use of series with negative powers for the analysis of singular points of solutions and monodromy.