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
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.