pendenze

Pendenze, also known as "pendencies" or "pendencies of a function," are a concept in mathematics, particularly in calculus and complex analysis. They refer to the set of points in the complex plane where a function fails to be analytic. In other words, pendenze are the points where the function does not possess a derivative, or where the derivative does not exist.

The term "pendenze" is derived from the Italian word for "slopes," reflecting the idea that the function's behavior at these points is not smooth or continuous. Pendenze can occur due to various reasons, such as discontinuities, poles, or essential singularities in the function.

In calculus, pendenze are often studied in the context of real-valued functions of a real variable. For example, the function f(x) = |x| has a pendenza at x = 0, as the function is not differentiable at this point.

In complex analysis, pendenze are studied in the context of complex-valued functions of a complex variable. For instance, the function f(z) = 1/z has a pendenza at z = 0, as the function has a pole at this point.

The study of pendenze is important in understanding the behavior of functions and their derivatives. By identifying the pendenze of a function, one can gain insights into the function's properties and its behavior in different regions of the complex plane.