Poles and Zeros

Introduction

In complex analysis, the behavior of a function is largely determined by its singularities—points where the function fails to be analytic. Among singularities, poles and zeros are the most tractable and important, forming the foundation for residue calculus and many applications.

A zero of a function ƒ is a point where ƒ(z)=0. A pole is a point where ƒ(z)→∞. These concepts are dual: zeros of ƒ correspond to poles of 1/ƒ, and vice versa. Understanding this duality illuminates the deep structure of complex functions.

The classification of singularities—as removable, poles, or essential—determines how functions can be extended and integrated. This chapter develops the theory of zeros and poles, establishes their connection to Laurent series, and explores their role in complex analysis.

Zeros of Analytic Functions

Definition and Order

Let ƒ be analytic in a neighborhood of (z_0). We say (z_0) is a zero of ƒ if ƒ((z_0))=0. The order (or multiplicity) of the zero is the smallest positive integer n such that ƒn*((z_0))≠0.

Equivalently, (z_0) is a zero of order n if ƒ can be written as:

ƒ(z)=(z−(z_0))n*g(z)

where g is analytic near (z_0) and g((z_0))≠0. The function g captures the non-vanishing part of ƒ near (z_0).

Taylor Series Characterization

Since ƒ is analytic, it has a Taylor series about (z_0):

ƒ(z)=(∑_k=0^∞)((a_k))*(z−(z_0))k

The zero has order n if and only if (a_0)=(a_1)=…=(a_n-1)=0 but (a_n)≠0. That is, the Taylor series begins:

ƒ(z)=(a_n)*(z−(z_0))n+(a_n+1)*(z−(z_0))(n+1)+⋯=(z−(z_0))n*((a_n)+(a_n+1)*(z−(z_0))+⋯)

The series in parentheses equals g(z), which is nonzero at z=(z_0) since g((z_0))=(a_n)≠0.

Isolation of Zeros

A fundamental property of analytic functions: zeros of a non-constant analytic function are isolated. There exists a neighborhood of each zero containing no other zeros.

This follows from the factorization ƒ(z)=(z−(z_0))n*g(z). Since g((z_0))≠0 and g is continuous, g is nonzero in some neighborhood of (z_0). Thus ƒ can only vanish at (z_0) in this neighborhood.

An important corollary: if ƒ is analytic on a connected domain and its zeros have an accumulation point in the domain, then ƒ is identically zero. This is the Identity Theorem.

Poles

Definition and Order

A function ƒ has a pole at (z_0) if ƒ is analytic in a punctured neighborhood 0<|z−(z_0)|<r but |ƒ(z)|→∞ as z→(z_0).

The order of the pole is the smallest positive integer n such that (z−(z_0))n*ƒ(z) has a removable singularity at (z_0) (can be extended to an analytic function with nonzero value at (z_0)).

Equivalently, ƒ has a pole of order n at (z_0) if:

ƒ(z)=h(z)/((z−(z_0))n)

where h is analytic near (z_0) and h((z_0))≠0.

Laurent Series at a Pole

Near a pole of order n, f has a Laurent series with finitely many negative powers:

ƒ(z)=(a_−n)/((z−(z_0))n)+(a_−n+1)/((z−(z_0))(n−1))+⋯+(a_−1)/(z−(z_0))+(a_0)+(a_1)*(z−(z_0))+⋯

The terms with negative powers form the principal part:

Principal part=(∑_k=1^n)((a_−k)/((z−(z_0))k))

The coefficient (a_1) is called the residue of ƒ at (z_0), denoted Res(ƒ,(z_0)). It plays a central role in contour integration.

Simple Poles

A pole of order 1 is called a simple pole. At a simple pole:

ƒ(z)=(a_−1)/(z−(z_0))+(a_0)+(a_1)*(z−(z_0))+⋯

The residue at a simple pole can be computed as:

Res(ƒ,(z_0))=(lim_z→(z_0))(z−(z_0))*ƒ(z)

If ƒ(z)=g(z)/h(z) where g((z_0))≠0 and h has a simple zero at (z_0), then:

Res(ƒ,(z_0))=g((z_0))/(h′*((z_0)))

Classification of Singularities

Isolated singularities come in three types, distinguished by their Laurent series.

Removable Singularities

A singularity is removable if ƒ can be extended to an analytic function at (z_0). Equivalently, the Laurent series has no negative powers (the principal part is zero).

Example: ƒ(z)=sin(z)/z has a removable singularity at z=0. The Taylor series of sin(z)=z−z3/6+… shows that sin(z)/z=1−z2/6+… is analytic at 0.

Poles

A pole has finitely many negative powers in its Laurent series. The order of the pole equals the highest power of 1/(z−(z_0)) appearing.

Poles are "controlled" singularities: the function blows up, but in a predictable way that can be cancelled by multiplying by (z−(z_0))n.

Essential Singularities

An essential singularity has infinitely many negative powers in its Laurent series. The behavior near an essential singularity is wild.

The Casorati-Weierstrass theorem: In any neighborhood of an essential singularity, ƒ takes values arbitrarily close to any complex number.

The stronger Picard theorem: In any neighborhood, ƒ actually takes every complex value infinitely often, with at most one exception.

Example: ƒ(z)=e(1/z) has an essential singularity at z=0. Its Laurent series is:

e(1/z)=1+1/z+1/(2!z2)+1/(3!z3)+⋯

with infinitely many negative powers.

The Argument Principle

The argument principle connects the zeros and poles inside a contour to an integral around the contour.

Let ƒ be meromorphic inside and on a simple closed contour C, with no zeros or poles on C. If Z counts zeros and P counts poles (with multiplicity), then:

1/(2*π*i)*(∮_C)((ƒ′*(z))/ƒ(z)*d(z))=Z−P

The integrand (ƒ′*(z))/ƒ(z)=d((log_)(ƒ(z)))/d(z) measures the change in the argument of ƒ(z) as z traverses C. Each zero inside C increases the winding number by 1; each pole decreases it by 1.

Connections to Other Topics

Residue Theorem

The residue theorem states that for a function meromorphic inside a contour C:

(∮_C)(ƒ(z)*d(z))=2*π*i*(∑_poles *(z_k)* inside *C^)(Res(ƒ,(z_k)))

The entire contribution to the contour integral comes from residues at poles. This transforms complex integrals into algebraic calculations.

Partial Fractions

A rational function can be decomposed into partial fractions, each associated with a pole. The coefficient of 1/((z−(z_0))k) in the partial fraction expansion relates directly to the Laurent series coefficients.

Meromorphic Functions

A function is meromorphic on a domain if its only singularities are poles. Meromorphic functions form a field under addition and multiplication. The quotient of any two analytic functions (with nonzero denominator) is meromorphic.

Summary

Zeros and poles are dual concepts characterizing the behavior of complex functions. A zero of order n satisfies ƒ(z)=(z−(z_0))n*g(z) with g((z_0))≠0. A pole of order n satisfies ƒ(z)=h(z)/((z−(z_0))n) with h((z_0))≠0.

Isolated singularities are classified by their Laurent series: removable singularities have no negative powers, poles have finitely many negative powers, and essential singularities have infinitely many.

The residue at a pole—the coefficient of 1/(z−(z_0)) in the Laurent series—is the key to the residue theorem. For simple poles,
Res(ƒ,(z_0))=(lim_z→(z_0))(z−(z_0))*ƒ(z).

The argument principle counts zeros minus poles via a contour integral. Zeros of analytic functions are isolated, leading to the Identity Theorem: if zeros accumulate, the function is identically zero.

These concepts form the foundation for residue calculus, which transforms complex integration into algebra, enabling the evaluation of difficult real integrals and the study of special functions throughout mathematics and physics.