Simple pole

Definition

A simple pole is a type of singularity for a complex function where the function approaches infinity as it approaches the pole, specifically when the pole has order one. At a simple pole, the function can be expressed in terms of its residue, and this property is crucial for evaluating integrals around the pole and applying residue calculus effectively.

5 Must Know Facts For Your Next Test

1. A simple pole occurs when the function behaves like \\$\frac{1}{z-a}\\$ near the pole at z = a.
2. The residue at a simple pole can be found using the formula \(\text{Res}(f, a) = \lim_{z \to a} (z - a) f(z)\).
3. Cauchy's integral formula allows for the evaluation of integrals of analytic functions around simple poles using their residues.
4. The contribution of a simple pole to an integral can be determined by multiplying the residue by 2πi, in accordance with the residue theorem.
5. Understanding simple poles is essential for mastering techniques like contour integration and calculating complex integrals.

Review Questions

• How does the concept of a simple pole relate to Cauchy's integral formula and its applications?
  • A simple pole is integral to Cauchy's integral formula because it allows us to compute integrals of functions that are analytic except at isolated singularities. When applying Cauchy's formula to a simple pole, we can directly use the residue at that pole to evaluate the integral. This connection showcases how residues can simplify complex evaluations, making it easier to analyze functions with singular behavior.
• Discuss how the residue theorem utilizes simple poles in evaluating contour integrals.
  • The residue theorem states that if you have a closed contour integral around isolated singularities, you can evaluate it by summing the residues of those singularities within the contour. For simple poles, calculating the residue is straightforward and crucial because it directly contributes to the value of the contour integral. Thus, understanding simple poles and their residues is essential for effectively applying the residue theorem in practical scenarios.
• Evaluate an integral involving a function with a simple pole at z = 2 over a closed contour surrounding this point. What steps would you take to solve this problem?
  • To evaluate an integral involving a function with a simple pole at z = 2, first identify the residue at this pole using \(\text{Res}(f, 2) = \lim_{z \to 2} (z - 2) f(z)\). Once you have calculated this residue, apply the residue theorem: multiply the residue by 2πi to find the value of the contour integral. This approach illustrates how knowing about simple poles allows us to handle complex integrals efficiently and accurately.

Related terms

Residue:

The residue of a complex function at a pole is the coefficient of the \\$\frac{1}{z-a}\\$ term in its Laurent series expansion around that pole.

Laurent Series: A Laurent series is a representation of a complex function as a series that includes terms with both positive and negative powers, useful for describing functions with singularities.

Contour Integration: Contour integration is a method of evaluating integrals along paths in the complex plane, particularly useful for functions with poles or other singularities.