Preface

The second edition of the book is published for several reasons. First, some new results appeared, while some required corrections and clarifications. After the publishing of the first edition, the book content has been studied by outside reviewers, some results and their following development have been reported at the conferences. So, the overall feedback from the readers and reviewers became such prominent that it required to be reflected in the material. In addition, new interesting results that originated on the basis of the previous material, have been discovered.
First of all, this relates to the discovery of relationships between the properties of polynomial, exponential, logarithmic and power functions. These new findings are such of both the theoretical and practical importance, that any material dealing with exponential functions has to include these results that substantially advance our understanding and knowledge of these important functions. (When we say, important”, we mean both the theoretical and practical importance of these new discoveries.)
Everybody who studies and uses the polynomial, exponential, logarithmic and power functions in any form will benefit from these new findings. (Power functions are similar to polynomials but they have real powers). If you work in any field that uses these functions in modeling of phenomena and approximating the behaviour of processes, such as physical, chemical, social, economical – actually these functions are extremely popular workhorses in all areas that use at least the very basic mathematical modeling – you will find such things that will make your understanding of modeling and your models much better.
The book presents the conceptual framework for modeling of natural phenomena. We show how the abstract mathematical concepts directly relate to the physical world and human societies; how to make the modeling adequate and meaningful, but not complicated. Nature has no boundaries, which we, humans, are trying to impose on it all the time, apparently in order to facilitate our studies. However, when the Whole is broken into too many pieces, the meaning of the Whole is lost. This is why we have to be so cautious dissecting the objects of our studies. Nature’s phenomena are inherently multifactor entities. This book uses this approach to study the subject.
These functions used to be considered as separate mathematical vehicles. In fact, their properties are interconnected, as we show this in the book. Relationships are important in all areas, they allow explaining and anticipating many things, help to avoid lots of troubles, while their knowledge helps to benefit in all phases.
How does one benefit from knowing the relationships between the aforementioned functions? We do not know a lot about each of these functions. However, if we know the relations between them, then everything we know about all functions can be applied to one function.

Chapters’ content at a glance

In chapter one, we describe the properties of each type of functions that we study. We start from polynomials. Then, we explore the power, exponential and logarithmic functions. Several illustrations and the study of functions’ asymptotic behaviour help to get an idea about functions’ specific features.
In chapter two, we introduce the properties and interrelationships of these functions, concentrating on new discoveries. In particular, we prove lemmas and theorems on how many real solutions the sum of exponential functions and power functions can have. We introduce the notions of corresponding functions and equations. For instance, each polynomial and power equation has a corresponding equation composed of the sum of exponential functions, so that the number of solutions of the polynomial and power equation is equal to the number of solutions of equation composed of the sum of exponential functions. In practical applications and modeling, this kind of knowledge is very valuable.
Chapter three studies the relations of Fermat’s Last Theorem with the physical world, similar to what we did in the first edition. However, the material is updated and new considerations and formulas have been added. Fermat’s Last Theorem, in fact, is also a particular case of exponential equation that is represented by a sum of exponential functions, although with integer variables. We begin the application of mathematical notions to the physical world and vice versa, that is extracting mathematical models from the physical reality. This is where we come to an understanding that the physical world is an inherently multifactor and interconnected entity that requires the appropriate multi-parameter models with interconnected parameters. How to do this? There is no simple answer, but we provide lots of examples and illustrations how to do this for the certain problems we studied. For instance, at this point, it should not be a mystery why in time of economical hardships in some foreign countries the outside aggressors and potential enemies begins to gain more attention in the headlines.
Chapter four is mathematical. Exponential functions, in many instances, adequately model the natural phenomena. In fact, exponential functions were discovered in Nature, they were not introduced as the purely mathematical concepts. Many fundamental Nature laws are based on exponential functions just because such is the nature of these phenomena, not because it happens that the exponential functions are slightly better than others modeling approaches. Exponential functions are not studied as good as, let us say, the polynomials. We introduced several Lemmas and mathematical notions which, in our view, enrich the theory of exponential functions with regard to their mathematical application to real phenomena.
In chapter five, we introduce several properties of sums of exponential functions and consider them in relation to modeling of natural phenomena. For instance, we consider oscillations in natural processes and show how the sums of exponential functions can adequately model this fairly typical behavioural scenarios, which often can be found in Nature.

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Last modified: 04/25/15