9.1 Singularities and poles

Complex functions can behave strangely at certain points. These points, called singularities, are where the function isn't smooth or well-defined. Understanding singularities is key to grasping complex analysis.

Poles are a special type of singularity where the function grows infinitely large. They're crucial for calculating integrals using the residue theorem, which is the main focus of this chapter on contour integration.

Types of Singularities

Isolated Singularities

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• Singularity a point where a complex function fails to be analytic (differentiable)
• Isolated singularity a singularity $z_0$ such that $f(z)$ is analytic in some deleted neighborhood of $z_0$
  • Deleted neighborhood excludes the point $z_0$ itself
• Three types of isolated singularities poles, removable singularities, and essential singularities
• Determining the type of isolated singularity involves examining the limit of $(z-z_0)^nf(z)$ as $z \to z_0$ for various values of $n$

Non-Isolated Singularities

• Branch point a type of non-isolated singularity
  • Occurs when a multi-valued function (square root, logarithm) is not analytic in any deleted neighborhood of the point
• Example of a branch point $f(z) = \sqrt{z}$ has a branch point at $z=0$
  • No matter how small a deleted neighborhood around $0$ is chosen, $f(z)$ will not be analytic due to the multiple values of the square root

Removable Singularities and Poles

• Removable singularity a singularity $z_0$ where the limit of $f(z)$ as $z \to z_0$ exists and is finite
  • The function can be redefined at $z_0$ to make it analytic there
  • Example $f(z) = \frac{\sin z}{z}$ has a removable singularity at $z=0$ since $\lim_{z \to 0} \frac{\sin z}{z} = 1$
• Pole a singularity $z_0$ where the limit of $|f(z)|$ as $z \to z_0$ is infinite
  • The function "blows up" to infinity near the pole
  • Example $f(z) = \frac{1}{z}$ has a pole at $z=0$ since $\lim_{z \to 0} |\frac{1}{z}| = \infty$

Essential Singularities

• Essential singularity a singularity that is neither removable nor a pole
  • The limit of $f(z)$ as $z \to z_0$ does not exist, even when allowing infinite values
• Characterized by highly erratic behavior near the singularity
  • The function values can approach any complex number or infinity in any neighborhood of $z_0$
• Example $f(z) = e^{\frac{1}{z}}$ has an essential singularity at $z=0$
  • As $z \to 0$, $f(z)$ oscillates wildly between very large and very small values

Pole Characteristics

Poles and Their Orders

• Pole a singularity $z_0$ where the limit of $|f(z)|$ as $z \to z_0$ is infinite
• Order of a pole the smallest positive integer $n$ such that $\lim_{z \to z_0} (z-z_0)^nf(z)$ is finite and nonzero
  • A pole of order $1$ is called a simple pole
  • Higher order poles ($n \geq 2$) are multiple poles
• Example $f(z) = \frac{1}{(z-1)^2}$ has a pole of order $2$ (double pole) at $z=1$
  • $\lim_{z \to 1} (z-1)^2f(z) = 1$, which is finite and nonzero

Laurent Series and Residues

• Laurent series an expansion of a complex function $f(z)$ in powers of $(z-z_0)$, valid in an annulus around $z_0$
  • Generalizes Taylor series to functions with singularities
  • $f(z) = \sum_{n=-\infty}^{\infty} a_n(z-z_0)^n$, where $a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz$
• Residue the coefficient $a_{-1}$ of the $\frac{1}{z-z_0}$ term in the Laurent series
  • Measures the "strength" of the singularity at $z_0$
  • For a pole of order $n$, the residue is given by $\frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} [(z-z_0)^nf(z)]$
• Example for $f(z) = \frac{1}{z^2}$, the Laurent series around $z=0$ is $\frac{1}{z^2} + 0 + 0 + \cdots$
  • The residue at $z=0$ is $0$, the coefficient of $\frac{1}{z}$