Conclusions and Outlook

We have shown that there is a close and deep relationship between real functions and complex analytic functions in the unit disk centered at the origin of the complex plane. This close relation between real functions and complex analytic functions allows one to use the powerful and extremely well-known machinery of complex analysis to deal with the real functions in a very robust way, even if the real functions are very far from being analytic. For example, the $\rho\to 1_{(-)}$ limit can be used to define the values of the functions or the values of their derivatives, at points where these quantities cannot be defined by purely real means. The concept of inner analytic functions played a central role in the analysis presented. The integral-differential chains of inner analytic functions, as well as the classification of singularities of these functions, which we introduced here, also played a significant role.

One does not usually associate non-differentiable, discontinuous and unbounded real functions with single analytic functions. Therefore, it may come as a bit of a surprise that all integrable real functions are given by the real parts of certain inner analytic functions on the open unit disk when one approaches the unit circle. Note, however, that there are many more inner analytic function within the open unit disk than those that were examined here, generated by integrable real functions. This leads to extensions of the relationship between inner analytic functions and real functions or related objects on the unit circle, which will be tackled in the aforementioned forthcoming papers.

One important limitation in the arguments presented here is that requiring that there be only a finite number of borderline hard singularities. It may be possible, perhaps, to lift this limitation, allowing for a denumerably infinite set of such integrable singularities. It is probably not possible, however, to allow for a densely distributed set of such singularities. Possibly, the limitation that the number of borderline hard singularities be finite may be exchanged for the limitation that the number of accumulation points of a denumerably infinite set of singular points with borderline hard singularities be finite.

It is quite apparent that the complex-analytic structure presented here can be used to discuss the Fourier series of real functions, as well as other aspects of the structure of the Fourier theory of real functions. The study of the convergence of Fourier series was, in fact, the way in which this structure was first unveiled. Parts of the arguments that were presented can be seen to connect to the Fourier theory, such as the role played by the Fourier coefficients, and the sufficiency of these Fourier coefficients to represent the functions, which relates to the question of the completeness of the Fourier basis of functions. This is a rather extensive discussion, which will be presented in a forthcoming paper.

It is interesting to note that the structure presented here may go some way towards explaining the rather remarkable fact that physicists usually operate with singular objects and divergent series in what may seem, from a mathematical perspective, a rather careless way, while very rarely getting into serious trouble while doing this. The fact that there is a robust underlying complex-analytic structure, that in fact explains how many such murky operations can in fact be rigorously justified, helps one to understand the unexpected success of this way to operate within the mathematics used in physics applications. In the parlance of physics, one may say that the complex-analytic structure within the unit disk functions as a universal regulator for all real functions, and related singular objects, which are of interest in physics applications.

We believe that the results presented here establish a new perspective for the analysis of real functions. The use of the theory of complex analytic functions makes it a rather deep and powerful point of view. Since complex analysis and analytic functions constitute such a powerful tool, with so many applications in almost all areas of mathematics and physics, it is to be hoped that other applications of the ideas explored here will in due time present themselves.