Analytical Functions

"from my book "functions and infinite series" www.mpantes.gr"---------------


the author      attempted  to expound the most useful facts about Taylor series of real and complex variables, as about Fourier transforms and Fourier series  in a mathematical narrative, in a semi-popular design. The book is not intended  for mathematics specialists, a reader is assumed whose mathematical  grounding  in other respects goes no further than the traditional second year University course in mathematics.

The Navier-Stokes differential equations describe the motion of fluids which are incompressible. The three-dimensional Navier-Stokes equations misbehave very badly although they are relatively simple-looking. The solutions could wind up being extremely unstable even with nice, smooth, reasonably harmless initial conditions. A mathematical understanding of the outrageous behaviour of these equations would dramatically alter the field of fluid mechanics. This paper describes why the three-dimensional Navier-Stokes equations are not solvable, i.e., the equations cannot be used to model turbulence, which is a three-dimensional phenomenon. [The paper is published in an international journal.]

In this paper, we investigate some argument properties for analytic functions with fixed second coefficient and positive real part. And we apply the argument properties to the functions that are analytic and normalized. In particular, the order of strongly starlikeness of strongly convex functions with fixed second coefficients is given.

We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if f(x) is small, then x is close to a zero of f. Working in Bishop's constructive setting, we first study pointwise versions of Z-stability and the related notion of good behaviour for functions. We then present a recursive counterexample to the classical argument for passing from pointwise Z-stability to a uniform version on compact metric spaces. In order to effect this passage constructively, we bring into play the positivity principle, equivalent to Brouwer's fan theorem for detachable bars, and the limited anti-Specker property, an intuitionistic counterpart to sequential compact-ness. The final section deals with connections between the limited anti-Specker property, positivity properties, and (potentially) Brouwer's fan theorem for detachable bars.