Book description
This book illustrates why abstract mathematical entities are needed to represent some aspects of physical reality. It provides an overview of different types of numbers and functions along with their historical background and applications.
Table of contents
 Cover Page
 Title Page
 Copyright Page
 Dedication
 Preface
 Contents
 1 On Different Types of Numbers

2 On e & ex
 2.1 Introduction
 2.2 Backdrop in which e emerged as the outcome of continuous compounding
 2.3 Outcome of decrease through continuous compounding
 2.4 e As an infinite series
 2.5 Proof of convergence of two sequences of e
 2.6 Proof of Irrationality of e
 2.7 Function ex
 2.8 Interesting features of exponential functions ex and emx
 2.9 Story of snails or an informal explanation of e1 visavis ei
 2.10 General function
 2.11 Infinite series representation of cos θ and sin θ
 2.12 Raising the power of e by complex angle (α + iθ)
 2.13 Rotating vector and concept of complex frequency
 2.14 cos θ and sin θ in Terms of exponential functions
 2.15 Obtaining an ellipse as the resultant of two rotating planar vectors rotating in opposite direction
 2.16 Problems related to division of a number
 2.17 Minimisation of the function xx
 2.18 Computation of ii
 2.19 Examples of 1st order differential equation
 2.20 Example of 2nd order differential equation
 2.21 Miscellaneous examples
 2.22 Matrix Exponential eA
 2.23 Chronology of development of concepts related to e
 Bibliography

3 Logarithm
 3.1 Introduction
 3.2 Logarithm as artificial numbers facilitating computation
 3.3 Logarithmic Function as an integral
 3.4 A story of the historical development of logarithm as an area
 3.5 Reverse Problem
 3.6 Some useful properties of logarithmic functions
 3.7 Expressing logarithm as a series
 3.8 Logarithmic curves
 3.9 LeibnitzGregory Series for π
 3.10 Schellbach's modified series for π
 3.12 Torricelli's Trumpet
 3.13 Logarithm of a complex number
 3.14 Resolving an apparent contradiction
 3.15 Applications
 3.16 Chronology of development of the concepts related to logarithm
 Bibliography

4 Concept of Complex Angle and Hyperbolic Functions
 4.1 Introduction
 4.2 Angle in terms of the area swept over during rotation
 4.3 Angle due to rotation from a vector view point and the concept of imaginary angle
 4.4 Angle due to stretching or shrinking and the concept of real angle
 4.5 Complex Angle
 4.6 Hyperbolic angle and Hyperbolic Functions for a hyperbola x2 – y2 = 1
 4.7 Area swept by a straight line joining the origin and a point moving over a hyperbola x2 – y2 = 1
 4.8 From the hyperbola of the form x2 – y2 = 1 to the hyperbola of the form u.v = 1
 4.9 Calculation of hyperbolic angle from the curve
 4.10 Calculation of traversed area while the tip of a straight line moves over the curvev v= 1/u
 4.11 Trigonometric functions of imaginary variable and Hyperbolic functions
 4.12 Trigonometric and Hyperbolic functions of complex angle α + iβ
 4.13 Applications
 4.14 Graphs of different hyperbolic functions
 4.15 Are the hyperbolic functions periodic?
 4.16 Expressions for inverse hyperbolic functions
 4.17 Infinite series representation of cosh x and sinh x
 4.18 Historical Development of the concept of hyperbolic functions
 Bibliography
 Index
Product information
 Title: A Journey into the World of Exponential Functions
 Author(s):
 Release date: June 2023
 Publisher(s): CRC Press
 ISBN: 9781000906226
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