Couth and uncouth function pairs

This article discusses the concept of 'couth' and 'uncouth' function pairs, coined by Robert Corless, in the context of inverting circular and hyperbolic trigonometric functions. It explains how naive inversion formulas may hold or fail depending on the definitions of inverse functions. Using Python's NumPy as an example, it shows that (sin, sinh) and (tan, tanh) are couth, while (cos, cosh) is uncouth. The article also examines reciprocal functions like cot, coth, sec, and csch, highlighting how different definitions of arccot can lead to couth or uncouth behavior. It provides code snippets for defining these functions and their inverses, and references a research paper from ACM SIGSAM Bulletin.