PART I: INTRODUCTION
1 Introduction
1.1 Introduction
1.2 Elementary Algebra
1.2.1 Quadratic polynomial
1.3 Finite Series
1.4 Infinite Series
1.4.1 Cauchy convergence
1.5 Problems
2 Functions
2.1 Introduction
2.2 Exponential function
2.3 Demand and supply function
2.4 Option theory payoff 2.5 Interest rates; bonds
2.6 Problems
PART II: LINEAR ALGEBRA 3 Simultaneous linear equations
3.1 Introduction
3.2 Two commodities
3.3 Vectors
3.4 Basis vectors
3.4.1 Scalar product
3.5 Linear transformations; matrices
3.6 EN: N-dimensional linear vector space
3.7 Linear transformations of EN
3.8 Problems
4 Matrices
4.1 Introduction
4.2 Matrix multiplication
4.3 Properties of N × N matrices
4.4 System of linear equations
4.5 Determinant: 2 × 2 case
4.6 Inverse of a 2 × 2 matrix
4.7 Outer product; transpose
4.7.1 Transpose
4.8 Eigenvalues and eigenvectors
4.8.1 Spectral decomposition
4.9 Problems
5 Square matrices
5.1 Determinant: 3 × 3 case
5.2 Properties of determinants
5.3 N × N determinant
5.3.1 Inverse of a N × N matrix
5.4 Leontief input-output model
5.4.1 Hawkins-Simon condition
5.5 Symmetric matrices
5.6 Symmetric matrix: diagonalization 5.6.1 Functions of a symmetric matrix
5.7 Hermitian matrices
5.8 Diagonalizable matrices
5.8.1 Non-symmetric matrix
5.9 Change of Basis states
5.9.1 Symmetric matrix: change of basis
5.9.2 Hermitian matrix: change of basis
5.10 Problems
PART III: CALCULUS
6 Integration 6.1 Introduction
6.2 Sums leading to integrals
6.3 Definite and indefinite integrals
6.4 Applications in economics
6.5 Multiple Integrals
6.5.1 Change of variables
6.6 Gaussian integration
6.6.1 N-dimensional Gaussian integration
6.7 Problems
7 Differentiation
7.1 Introduction
7.2 Inverse of Integration
7.3 Rules of differentiation 7.4 Integration by parts
7.5 Taylor expansion
7.6 Minimum and maximum
7.6.1 Maximizing profit 7.7 Integration; change of variable
7.8 Partial derivatives
7.8.1 Chain rule; Jacobian
7.8.2 Polar coordinates; Gaussian integration 7.9 Hessian matrix: critical points
7.10 Constrained optimization: Lagrange multiplier
7.10.1 Interpretation of λc
7.11 Line integral; Exact and inexact differentials
7.12 Problems
8 Functional analysis
8.1 Dirac bracket and vector notation
8.2 Continuous basis states
8.3 Dirac delta function
8.4 Basis states for function space
8.5 Operators on function space
8.6 Gaussian kernel
8.7 Fourier Transform
8.8 Taylor expansion
8.9 Gaussian functional integration
8.10 Problems
9 Ordinary Differential Equations
9.1 Introduction
9.2 Separable differential equations 9.3 Linear differential equations
9.4 Bernoulli differential equation
9.5 Homegeneous differential equation
9.6 Second order linear differential equations
9.6.1 Single eigenvalue
9.7 Ricatti differential equation
9.8 Inhomogeneous second order differential equations
9.8.1 Green's function
9.9 System of linear differential equations
9.10 Strum-Louisville theorem; special f
About the Author:
Prof. Belal Ehsan Baaquie holds a B.S. in Physics from Caltech and a Ph.D. in Theoretical Physics from Cornell University, USA. His main research interest is in the study and application of mathematical methods from quantum field theory. He has applied the mathematical formalism of field theory to finance and been a major contributor to the emerging field of quantum finance. His current focus is on developing the formalism of quantum finance and applying it to option pricing, corporate coupon bonds, and the theory of interest rates, as well as the study of equity, foreign exchange, and commodities. He is also applying methodologies from statistical mechanics and quantum field theory to the study of microeconomics and macroeconomics.
