ABSTRACT
The exponential is the simplest integral function that can be defined (Subsection 3.3.3) in six equivalent ways: (D1) by being equal to its own derivative (Sections 1.1.8 and 3.1); (D2) by a power series (Sections 1.1.8 and 3.1); (D3) by the limit of a binomial (Section 3.1); (D4) by one of a number of equivalent continued fractions (Section 3.2); (D5) by the property of changing sums to products (Subsection 3.3.1); and (D6) by the property of changing products to powers (Subsection 3.3.2). For unicity, the definition D1 (D5 and D6) requires that the function (its derivative) be unity at the origin. Of the representations (D2, D3, and D4), those that lead to numerical computation with faster convergence are the series (D2) and the continued fraction (D3); the latter also proves that the number e is irrational. Any of these definitions leads to (Section 3.4) the other properties of the exponential: (1) it has an essential singularity at infinity, and so grows faster than any power; (2) it has no zeros (poles) and hence no representation as infinite product (series of fractions); (3) the latter property can be extended to show that any integral function without zeros is the exponential of another integral function. The exponential is periodic, hence many-valent, and thus, its inverse (Section 3.5), the logarithm, is many-valued and becomes single-valued in the complex plane with a branch-cut (Sections I.7.2 and I.7.3). The existence of a branchpoint limits the region of convergence of the power series for the logarithm, that can also be represented by continued fractions (Section 3.6). The logarithm transforms products to sums, and powers to products, thus owing its usefulness to this simplification of operations. The exponential and its inverse, the natural logarithm, allow the definition of powers and logarithms with arbitrary base (Section 3.7), that is, other than the number e. All these functions can be calculated from logarithmic tables (Section 3.8), whose computation can be performed using series expansions (Section 3.6) combined with elementary properties. Some of the continued fractions for elementary transcendental functions, such as the exponential (logarithm) in spite of their apparent simplicity [Section 3.2 (Section 3.6)], require for proof the Lambert method (Section 1.9), involving higher transcendental functions (Section 3.9) such as the confluent (Gaussian) hypergeometric series.
