One Raised to the Infinite Power, the Concept of Natural Logarithms, and Hyperbolic Trigonometry

This chapter returns the overall focus of this treatise to the development of those unique attributes of calculus begun in Chapters 3 and 4; namely differentiation and integration. Having diverted attention in order to follow what was a seemingly tangential path of analytic geometry in Chapter 5, the main stream of calculus is returned to with a third indeterminate combination manifesting some unique attributes of major scientific significance. Much as the number pi was discovered four millennia ago, another constant irrational number, discovered much more recently, is associated with that particular integral for which the formula for integration of polynomials could not be extended; namely, when the exponent n had the value of −1.

The physically important constant number discovered by John Napier and designated as e in honor of Leonhart Euler [23], along with its associated function ex, are next examined. Instead of the traditional definitions given in modern calculus textbooks, such as [14] through [21], this treatise proposes as the source of definition that binary indeterminate form 1∞. This constant will be associated with properties common to logarithms in algebra. For such a development one reiterates the denotation and connotation of the terms “exponent” and “logarithm”; two terms which are essentially synonymous in denotation but whose connotations are that “exponent” is usually associated with an integer or a common fraction, while “logarithm” is traditionally a decimal. In Chapter 2, the properties of logarithms were developed for any base, but primarily when the base of the numbering system was the number 10. Such logarithms were designated with the adjective “common” in conformity with the basis of our number system being the biological “accident” that the human species has evolved to have ten fingers.

The definition of e will be in terms of a definite integral with the argument of the function being one, or both, of its limits and the integrand being a “dummy variable”. The added perspective that will accompany this fundamental constant of both nature and of advanced mathematics, e, will be a third member of the set of indeterminate forms, cataloged by l'Hôpital in 1696, which was discussed, but not then assigned a name other than L7 in Section 3.2; namely one raised to the infinite power (1∞).

Next, employing algebraic properties associated with exponents, a pragmatic technique, which is applicable to functions that are combinations of multiplication, division, exponents and roots, while simultaneously being limited with respect to addition and subtraction will be described. This technique, called logarithmic differentiation, has been devised so as to decrease the tedium of selected traditional differentiations in many instances by employing properties of algebra.

In a similar manner, this number e, as the base of the function ex, has the important functional identity property that its derivative is equal to the function itself. Moreover, every higher derivative (and integral, when the constants of integration are set equal to zero) is also equal to this function. It is now further observed that the sum and difference of this function and its reciprocal bear a seemingly serendipitous relation to the respective cosine and sine functions of trigonometry. It is precisely one-half of each of these two relations which have been the traditional nearly-universal definition of that group of functions referred to as the hyperbolic trigonometric functions; i.e., cosh x = (ex + e-x)/2 and sinh x = (ex − e-x)/2. To the contrary, this treatise defines such a set of functions, as their names imply, starting from the geometry of a selected reference hyperbola. By the proposed geometric definitions, all of the identity properties of these functions are derivable without reference to their relation to exponential functions. In other word, the exponential relationships are downgraded to being secondary properties, while the trigonometry of the reference hyperbola is elevated to being the basis for definition. That the exponential relations are, in fact, valid is an intrinsic property of the confluence of algebra and geometry. This will be shown in Chapter 7 by expanding each of these functions using infinite series. Consequently, each is a different path to the same mathematical description in much the same manner as the set of six fundamental functions in circular trigonometry was defined by starting from either angles in a right triangle or lengths in a unit circle.

Keywords: Exponential Function ex, Hyperbolic Trigonometry, Derivation from Hyperbola vs. Exponential Functions, Logarithmic Differentiation, Naperian Constant e, Natural Logarithms