The discussion clarifies that "reduction formulas" and "recursive formulas" in the context of integration by parts refer to the same concept, specifically in evaluating trigonometric integrals. Reduction formulas involve expressing integrals of higher degrees in terms of integrals of lower degrees, similar to how recursive definitions operate. The example provided is the factorial function, where n! = (n - 1)!n illustrates the recursive nature. Ultimately, both terms describe a method for simplifying integrals through a systematic approach.
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voko said:Recursive means something defined in terms of itself, but with a different conditions. For example, factorial: n! = (n - 1)!n.
It is not entirely clear what in you book is called "reduction formulas", but if that means, for example, something with degrees defined in terms of something similar with smaller degrees, then this could be equally called recursive formulas.