Simplified Derivation of the Properties of Elementary Transcendentals

Abstract

The purpose of the present paper is to give a derivation of the properties of the elementary transcendentals from a more unified point of view than is customary. It will be assumed that the general properties of analytic functions are established. In particular we shall use the theorem that the real (or imaginary) part of an analytic function cannot assume its maximum value in the inside of a closed domain in which it is regular. Similarly, use will be made of Weierstrass’s theorem about a set of uniformly converging analytic functions, of Liouville’s and Rolle’s theorems.§ All of these theorems can be established without the use of the properties of the transcendental functions, using only elementary calculations and the concept of convergence. The present paper shows how the properties of the elementary transcendentals can then be derived from the aforementioned theorems by means of the methods of the theory of complex functions. From the point of view of this theory, this appears to be more appropriate than taking over the results concerning the properties of these functions, at least for real arguments, from other parts of analysis. An outline of our procedure follows.