Residue at a pole

Definition

The residue at a pole is a complex number that represents the coefficient of the $(z - z_0)^{-1}$ term in the Laurent series expansion of a function around that pole, where $z_0$ is the location of the pole. This concept is essential in evaluating complex integrals and plays a critical role in both Cauchy's integral formula and the residue theorem, providing a method to compute contour integrals by relating them to residues of singularities.

5 Must Know Facts For Your Next Test

1. Residues can be computed using limits, specifically the formula: $$ ext{Res}(f, z_0) = rac{1}{(m-1)!} rac{d^{m-1}}{dz^{m-1}} igg|_{z=z_0} (z - z_0)^m f(z)$$ for a pole of order $m$.
2. Residues are used to evaluate complex integrals through the residue theorem, which states that the integral of a function around a closed contour equals $2 ext{i}\pi$ times the sum of the residues inside that contour.
3. If a function has isolated singularities, the residues at these poles will determine the behavior of the integral along paths that enclose these singularities.
4. Residues are not just limited to poles; they can also be computed at essential singularities, but the calculation methods differ.
5. The residue at a simple pole (first-order pole) can be calculated directly as $$ ext{Res}(f, z_0) = rac{g(z_0)}{h'(z_0)}$$ if $f(z) = rac{g(z)}{h(z)}$ and $h(z_0) = 0$ with $h'(z_0) \neq 0$.

Review Questions

• How do you compute the residue at a pole using limits, and why is this method significant?
  • To compute the residue at a pole, you can use the limit formula: $$ ext{Res}(f, z_0) = rac{1}{(m-1)!} rac{d^{m-1}}{dz^{m-1}} igg|_{z=z_0} (z - z_0)^m f(z)$$ for a pole of order $m$. This method is significant because it provides a systematic way to find residues without needing to derive the full Laurent series. Understanding how to compute residues this way is crucial for applying Cauchy's integral formula and residue theorem effectively.
• Explain how residues at poles are utilized in the residue theorem to evaluate integrals.
  • In the residue theorem, if you have a function with isolated singularities within a closed contour, you can evaluate the contour integral by summing the residues at these poles. Specifically, the theorem states that if $f(z)$ is analytic inside and on some closed contour $C$, except for a finite number of poles, then $$\oint_C f(z) dz = 2\pi i \sum_{k} ext{Res}(f, z_k)$$ where $z_k$ are the poles inside $C$. This powerful result allows us to calculate complex integrals simply by focusing on singularities rather than evaluating the integral directly.
• Analyze how understanding residues at poles enhances your ability to solve complex integral problems in various scenarios.
  • Understanding residues at poles enhances your ability to solve complex integrals because it simplifies the evaluation process for contours enclosing singularities. By knowing how to identify and calculate residues, you can efficiently apply Cauchy's integral formula and residue theorem in diverse situations, such as in physics or engineering problems involving wave functions or electric fields. This knowledge enables you to tackle challenging problems by converting potentially complex integral evaluations into manageable calculations based on residues.

Related terms

Laurent series: A representation of a complex function that includes terms with negative powers, used to express functions around singular points.

Cauchy's integral formula: A fundamental result in complex analysis that provides an explicit formula for the values of holomorphic functions based on contour integrals around points.

pole: A type of singularity where a function approaches infinity, typically characterized by having a finite order.