shortly we can derive the Cauchy's residue theorem (its general form) just by direct integration of a Taylor series placing an open curve onto a specific domain, in order to satisfy holomorphic properties
The problem of fraction decomposition it's easy to solve by using the cover up method, when there are no repeated linear factors in the denominator . Nevertheless it could turn into a hard work if these factors are raised to a high power, where the cover up method doesn't work . This technique shows how to calculate these coefficients without solving large systems of equations with a clever rearrangement of the numerator.
The problem of integration technique over integrands of the form f(t)/t^n, can be solved by differentiation(n times) by using Leibniz's rule to get rid of t^n, that leads to integrate back (n times) to end the game which it's harder than the original problem.This work focuses on the derivation of the formula (Pagano's Theorem) which is a perfect tool to avoid that hard task. It allows to change the difficult n iterated integrals into a more outstanding easier problem which consists of n -...
