F(z) = 1/(z^2 + 1)

Definition

The function f(z) = 1/(z^2 + 1) is a complex function that has singularities at points where the denominator equals zero, specifically at z = i and z = -i. This function is significant because it serves as a classic example for applying the residue theorem to compute complex integrals around contours that enclose these singularities.

5 Must Know Facts For Your Next Test

1. The singularities of f(z) are located at z = i and z = -i, both of which are simple poles since they result from linear factors in the denominator.
2. To apply the residue theorem, one typically encloses one or both poles within a closed contour, such as a circle of radius R centered at the origin.
3. The residues at the poles can be computed using the formula for simple poles, which involves finding the limit of (z - pole) * f(z) as z approaches the pole.
4. The integral of f(z) around a closed contour that includes both poles can be evaluated using the residue theorem, resulting in 2πi times the sum of residues at those poles.
5. The residue at each pole can provide insights into the behavior of f(z) in its vicinity, allowing us to understand how f(z) contributes to integrals over certain paths in the complex plane.

Review Questions

• How does the presence of singularities in f(z) affect the evaluation of contour integrals?
  • The presence of singularities like those found in f(z) significantly impacts how contour integrals are evaluated. When integrating around these points, we can use the residue theorem, which allows us to compute the integral based on the residues at these singularities. This technique simplifies what would otherwise be complicated calculations and provides a powerful tool for understanding the function's behavior near these critical points.
• In what way does calculating residues at the poles of f(z) contribute to understanding complex integrals?
  • Calculating residues at the poles of f(z) directly contributes to our ability to evaluate complex integrals by providing specific values that can be used in conjunction with the residue theorem. The residues encapsulate key information about how f(z) behaves near its singularities. When we sum these residues and multiply by 2πi, we obtain precise results for integrals over contours enclosing these poles, highlighting their importance in complex analysis.
• Evaluate the integral of f(z) along a closed contour that encloses both poles and discuss its implications.
  • To evaluate the integral of f(z) along a closed contour that encloses both poles (z = i and z = -i), we first calculate the residues at each pole. For z = i, we find that the residue is 1/2i, and for z = -i, it is -1/2i. The sum of these residues is zero. Therefore, applying the residue theorem indicates that the integral around this contour equals 0. This result reflects Cauchy's integral theorem, showing that if a function is analytic inside and on a closed contour except for isolated singularities, the integral can still yield meaningful conclusions about its overall behavior.

Related terms

Residue:

A residue is a complex number that represents the coefficient of (z-a)^{-1} in the Laurent series expansion of a function around a singularity a, playing a critical role in evaluating integrals using the residue theorem.

Singularity:

A point at which a complex function ceases to be analytic; for f(z), the singularities occur where z^2 + 1 = 0.

Contour Integral:

An integral taken over a path or contour in the complex plane, often used in conjunction with the residue theorem to evaluate integrals involving complex functions.