Calculating Residues
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a−1 of (z − c)−1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
According to Cauchy's integral formula, we have:
where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle of radius ε around c, where ε is as small as we desire. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.
Read more about this topic: Residue (complex Analysis)
Famous quotes containing the words calculating and/or residues:
“[The] elderly and timid single gentleman in Paris ... never drove down the Champs Elysees without expecting an accident, and commonly witnessing one; or found himself in the neighborhood of an official without calculating the chances of a bomb. So long as the rates of progress held good, these bombs would double in force and number every ten years.”
—Henry Brooks Adams (1838–1918)
“You were born into a different world that will present you with different gifts and challenges. A new vision of manhood will be called for that does not tie so closely into the more aggressive and competitive residues of our male character. You will need to search out new ways of expressing strength, showing mastery, and exhibiting courage—ways that do not depend upon confronting the world before you as an adversary.”
—Kent Nerburn (20th century)