The discussion revolves around the concept of independence between functions, specifically addressing when two functions, y1 = f1(x) and y2 = f2(x), can be considered independent or dependent on each other. The scope includes theoretical considerations and definitions related to function independence, linear independence, and the implications of the Wronskian determinant.
Participants express differing views on the concept of independence among functions, with some arguing for a general dependency and others emphasizing the distinction of linear independence. The discussion remains unresolved regarding the precise definitions and implications of function independence.
There is a lack of clarity regarding the definitions of independence being used, and participants highlight the need for a more rigorous framework to discuss the relationships between functions. The discussion also touches on the mathematical properties of the Wronskian, which may not be fully understood by all participants.
Ben Niehoff said:We can, however, talk about linearly-independent functions: namely, functions with vanishing Wronskian.
I have no idea what you are saying! Do y1, y2, x1, x2 just represent numbers? What, then, is the difference between saying y1= f1(x1) and just y= f1(x)? And, of course, what do you mean by "independent"? Apparently you don't mean "linear independence". Before anyone can tell you whether or not "any two functions are not independent" you will have to say what you mean by two functions being "independent"!Heirot said:When are to functions y1 = f1(x) and y2 = f2(x) independent? It would apper never, because, we can always write x = f1-1 (y1), and therefore y2 is a function of y1. Every function is dependent of any other function. Generally, dy1/dy2 != 0 for arbitrary functions y1 and y2. Is this reasoning correct?