Searching for the ideal definition of what a function is is an Eulerian mathematical crusade of the 21st century. This is the Holy Grail of mathematics as what a TOE is to Physics and the cup of the Last Supper is to Christianity.
As noted by Roger Penrose in his book ‘The Road to Reality’, 2004 there are two approaches: (1) the real one by power series expansion and (2) the imaginary one by complex analysis. However, as also pointed out by Penrose, at the local infinitesimal domain of space-time, these are equivalent. Nevertheless within the real, there exist two binary operations. These are the operation of addition and multiplication.
A function f(c), by the real approach is represent able by its power series expansion: f(c)=a0+a1c+a2c²+a3c³+a4c+… . The necessary condition for the existence all these a-coefficients is that the function itself must be differentiable or the same as saying that for each step of differentiation a limit exists. The result is Maclaurin’s series and its analyticity.


LinkBack URL
About LinkBacks
Reply With Quote


