Zeros and poles are crucial concepts in complex analysis, revealing key behaviors of analytic functions. They help us understand where functions vanish or become unbounded, providing insights into their overall structure and properties.

Exploring zeros and poles connects to the broader study of series representations. By analyzing these special points, we can develop Laurent series expansions, which generalize Taylor series and allow us to represent functions with singularities.

Zeros and Poles of Functions

Defining Zeros and Poles

• A zero of an analytic function $f(z)$ is a complex number $z_0$ that satisfies the equation $f(z_0) = 0$
• A pole of an analytic function $f(z)$ is a complex number $z_0$ where the function becomes unbounded (approaches infinity) as $z$ approaches $z_0$
• Zeros and poles are isolated singularities of analytic functions (points where the function is not analytic)
• If an analytic function $f(z)$ has a zero at $z_0$, then $z_0$ is a root of the equation $f(z) = 0$ ($e^z - 1 = 0$ has a zero at $z_0 = 0$)
• If an analytic function $f(z)$ has a pole at $z_0$, then $z_0$ is a root of the equation $1/f(z) = 0$ ($1/(z^2 - 1)$ has poles at $z_0 = \pm 1$)

Finding Zeros and Poles of Rational Functions

• The zeros and poles of a rational function (a function that can be written as the ratio of two polynomials) can be found by factoring the numerator and denominator
  • The zeros are the roots of the numerator polynomial
  • The poles are the roots of the denominator polynomial
• Example: For the rational function $f(z) = (z - 1)(z + 2)/(z - 3)(z + 4)$
  • Zeros: $z_0 = 1$ and $z_0 = -2$
  • Poles: $z_0 = 3$ and $z_0 = -4$

Order of Zeros and Poles

Order of Zeros

• The order (or multiplicity) of a zero $z_0$ of an analytic function $f(z)$ is the smallest positive integer $m$ such that the $m$th derivative of $f(z)$ at $z_0$ is non-zero
  • $f(z_0) = f'(z_0) = ... = f^{(m-1)}(z_0) = 0$ and $f^{(m)}(z_0) \neq 0$
• If $f(z)$ has a zero of order $m$ at $z_0$, then $f(z)$ can be written as $f(z) = (z - z_0)^m g(z)$, where $g(z)$ is analytic and non-zero at $z_0$
• Example: $f(z) = (z - 1)^2(z + 2)$ has a zero of order 2 at $z_0 = 1$ and a zero of order 1 at $z_0 = -2$

Order of Poles

• The order (or multiplicity) of a pole $z_0$ of an analytic function $f(z)$ is the smallest positive integer $n$ such that the limit of $(z - z_0)^n f(z)$ as $z$ approaches $z_0$ is non-zero and finite
• If $f(z)$ has a pole of order $n$ at $z_0$, then $f(z)$ can be written as $f(z) = (z - z_0)^{-n} h(z)$, where $h(z)$ is analytic and non-zero at $z_0$
• Example: $f(z) = 1/(z - 3)^2(z + 4)$ has a pole of order 2 at $z_0 = 3$ and a pole of order 1 at $z_0 = -4$

Function Behavior Near Singularities

Types of Isolated Singularities

• Isolated singularities are classified as removable singularities, poles, or essential singularities based on the behavior of the function near the singularity
• A removable singularity is a point $z_0$ where the function is undefined, but the limit of the function as $z$ approaches $z_0$ exists and is finite
  • The function can be redefined at $z_0$ to make it analytic ($f(z) = (z^2 - 1)/(z - 1)$ has a removable singularity at $z_0 = 1$)
• A pole is a point $z_0$ where the function becomes unbounded as $z$ approaches $z_0$
  • The order of the pole determines the rate at which the function grows ($1/(z - 3)^2$ has a pole of order 2 at $z_0 = 3$)
• An essential singularity is a point $z_0$ where the function exhibits complicated behavior as $z$ approaches $z_0$, and the limit does not exist or is infinite
  • The function cannot be defined at $z_0$ to make it analytic ($e^{1/z}$ has an essential singularity at $z_0 = 0$)

Casorati-Weierstrass Theorem

• The Casorati-Weierstrass theorem states that in any neighborhood of an essential singularity, an analytic function takes on all possible complex values, with at most one exception, infinitely often
• This theorem illustrates the complex behavior of functions near essential singularities
• Example: In any neighborhood of the essential singularity at $z_0 = 0$ for $f(z) = e^{1/z}$, the function takes on all possible complex values, except possibly one value, infinitely often

Laurent Series Expansions for Poles

Laurent Series Definition

• A Laurent series is a generalization of a Taylor series that allows for negative powers of $(z - z_0)$ and is used to represent functions with poles
• The Laurent series of a function $f(z)$ centered at $z_0$ is given by $f(z) = \sum_{n=0}^{\infty} a_n(z - z_0)^n + \sum_{n=1}^{\infty} b_n(z - z_0)^{-n}$, where $a_n$ and $b_n$ are complex coefficients
• The principal part of the Laurent series consists of the terms with negative powers of $(z - z_0)$ and represents the behavior of the function near the pole
• Example: The Laurent series expansion of $f(z) = 1/(z - 1)^2$ centered at $z_0 = 1$ is $f(z) = (z - 1)^{-2} + 0 + 0 + ...$

Residues and the Residue Theorem

• The residue of a function $f(z)$ at a pole $z_0$ is the coefficient $b_1$ of the $(z - z_0)^{-1}$ term in the Laurent series expansion
• The residue can be calculated using the formula: $\text{Res}[f(z), z_0] = \frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}}[(z - z_0)^n f(z)]$, where $n$ is the order of the pole
• The residue theorem relates the residues of a function to the integral of the function along a closed contour and is a powerful tool for evaluating complex integrals
• Example: For $f(z) = 1/(z^2 - 1)$, the residues at the poles $z_0 = \pm 1$ are $\text{Res}[f(z), 1] = 1/2$ and $\text{Res}[f(z), -1] = -1/2$