Chapter 11 - Transcendental Functions

Abstract

Chapter 11 discusses transcendental functions. These functions can be obtained as a sum of functional series and improper integrals, depending on a parameter. Logarithmic and exponential, trigonometric and hyperbolic functions are defined in these ways and their properties are deduced. Power series are discussed and analytic functions are defined. An interesting feature of this chapter is an introduction to multiplicative calculus, which presents a calculus alternative to calculus of Newton and Leibnitz. By use of methods of multiplicative calculus it is proved that an infinitely many times differentiable function may not be analytic. Infinite products are handled. Improper integrals, depending on a parameter, are discussed and used to define Euler’s Gamma and Beta functions.