A Journey into the World of Exponential Functions

Table of contents

1. Cover Page
2. Title Page
3. Copyright Page
4. Dedication
5. Preface
6. Contents
7. 1 On Different Types of Numbers
   1. 1.1 The beginning of the journey
   2. 1.2 Extension of Number Field
      1. 1.2.1 Negative Number
      2. 1.2.2 Fraction and Rational Number
      3. 1.2.3 Irrational Number
      4. 1.2.4 Imaginary (or Quadrature Number)
      5. 1.2.5 Quaternion
      6. 1.2.6 Infinitesimal and Hyper-real Numbers
   3. Bibliography
8. 2 On e & ex
   1. 2.1 Introduction
   2. 2.2 Backdrop in which e emerged as the outcome of continuous compounding
   3. 2.3 Outcome of decrease through continuous compounding
   4. 2.4 e As an infinite series
   5. 2.5 Proof of convergence of two sequences of e
   6. 2.6 Proof of Irrationality of e
   7. 2.7 Function ex
   8. 2.8 Interesting features of exponential functions ex and emx
   9. 2.9 Story of snails or an informal explanation of e1 vis-a-vis ei
      1. 2.9.1 e as a distance travelled along a road
      2. 2.9.2 ei as a Distance travelled in afield
   10. 2.10 General function
   11. 2.11 Infinite series representation of cos θ and sin θ
   12. 2.12 Raising the power of e by complex angle (α + iθ)
   13. 2.13 Rotating vector and concept of complex frequency
   14. 2.14 cos θ and sin θ in Terms of exponential functions
   15. 2.15 Obtaining an ellipse as the resultant of two rotating planar vectors rotating in opposite direction
   16. 2.16 Problems related to division of a number
       1. 2.16.1 Maximisation of the function
       2. 2.16.2 Dividing a number into several equal parts such that their product is maximum
   17. 2.17 Minimisation of the function xx
   18. 2.18 Computation of ii
   19. 2.19 Examples of 1st order differential equation
   20. 2.20 Example of 2nd order differential equation
   21. 2.21 Miscellaneous examples
       1. 2.21.1 Radioactive Disintegration
       2. 2.21.2 Advance of Chemical Reaction
       3. 2.21.3 Logistic growth
   22. 2.22 Matrix Exponential eA
   23. 2.23 Chronology of development of concepts related to e
   24. Bibliography
9. 3 Logarithm
   1. 3.1 Introduction
   2. 3.2 Logarithm as artificial numbers facilitating computation
   3. 3.3 Logarithmic Function as an integral
   4. 3.4 A story of the historical development of logarithm as an area
   5. 3.5 Reverse Problem
   6. 3.6 Some useful properties of logarithmic functions
   7. 3.7 Expressing logarithm as a series
      1. 3.7.1 Alternate Method of Derivation of Logarithmic Series
      2. 3.7.2 Searching for a method with a higher rate of convergence
      3. 3.7.3 Further improvement of the method described in section 3.7.2
   8. 3.8 Logarithmic curves
      1. 3.8.1 Observations on the curve for the functiony = ln x
      2. 3.8.2 Curve for the function y = ln(–x) and y = ln |x|
   9. 3.9 Leibnitz-Gregory Series for π
   10. 3.10 Schellbach's modified series for π
   11. 3.12 Torricelli's Trumpet
   12. 3.13 Logarithm of a complex number
   13. 3.14 Resolving an apparent contradiction
   14. 3.15 Applications
       1. 3.15.1 Inductance of a two wire line:
       2. 3.15.2 Capacitance of a two wire line
       3. 3.15.3 Loudness of sound
       4. 3.15.4 Magnitude of earthquake
       5. 3.15.5 Acidity of a substance
   15. 3.16 Chronology of development of the concepts related to logarithm
   16. Bibliography
10. 4 Concept of Complex Angle and Hyperbolic Functions
    1. 4.1 Introduction
    2. 4.2 Angle in terms of the area swept over during rotation
    3. 4.3 Angle due to rotation from a vector view point and the concept of imaginary angle
    4. 4.4 Angle due to stretching or shrinking and the concept of real angle
    5. 4.5 Complex Angle
    6. 4.6 Hyperbolic angle and Hyperbolic Functions for a hyperbola x2 – y2 = 1
    7. 4.7 Area swept by a straight line joining the origin and a point moving over a hyperbola x2 – y2 = 1
    8. 4.8 From the hyperbola of the form x2 – y2 = 1 to the hyperbola of the form u.v = 1
    9. 4.9 Calculation of hyperbolic angle from the curve
    10. 4.10 Calculation of traversed area while the tip of a straight line moves over the curvev v= 1/u
    11. 4.11 Trigonometric functions of imaginary variable and Hyperbolic functions
    12. 4.12 Trigonometric and Hyperbolic functions of complex angle α + iβ
    13. 4.13 Applications
        1. 4.13.1 A DC Voltage Source (V) is switched on to an LC circuit
        2. 4.13.2 A DC Voltage Source is switched on to a RLC circuit
        3. 4.13.3 Catenary
        4. 4.13.4 A Leaky Direct Current Line
        5. 4.13.5 Dynamics of moving bodies
    14. 4.14 Graphs of different hyperbolic functions
    15. 4.15 Are the hyperbolic functions periodic?
    16. 4.16 Expressions for inverse hyperbolic functions
    17. 4.17 Infinite series representation of cosh x and sinh x
    18. 4.18 Historical Development of the concept of hyperbolic functions
    19. Bibliography
11. Index