Numbers and Functions: Theory, Formulation, and Python Codes

Contents
About the Author
1. Introduction
	1.1 Computational methods
	1.2 Why contribute to this book series
	1.3 Two most fundamental concepts: Numbers and functions
	1.4 Who may read this book
	1.5 Codes used in this book
	1.6 Use of external modules or dependencies
	1.7 Use of help()
	Reference
2. Real Numbers
	2.1 The set of real number R
	2.2 Limits and accuracy of a real number in Python
		2.2.1 Limits for integers in Python
		2.2.2 Limits and accuracy for float real numbers
	2.3 Examples: Real numbers produced using Python
	2.4 Random sampling of numbers over a domain
	2.5 Controlled random sampling
	2.6 Concept of closure
	2.7 Numpy code for the closure test of real numbers
	2.8 Normalization or scaling
	2.9 R is closed under the arithmetic operations
	2.10 Norm of a real number
	2.11 Real coordinate spaces
		2.11.1 Numbers in one-dimensional real coordinate space R1
		2.11.2 Vectors in 2D real coordinate space R2
		2.11.3 Vectors in n-dimensional real coordinate space Rn
	2.12 Remarks
	References
3. Complex Numbers
	3.1 Definition
	3.2 Geometrical representation
		3.2.1 Re∼Im plane
		3.2.2 Periodicity of complex numbers
		3.2.3 Euler’s equation
		3.2.4 Multi-values of numbers in Re∼Im plane
	3.3 Arithmetic operations for complex numbers
	3.4 Conjugate of a complex number
	3.5 Norm of a complex number
	3.6 Numerical test on the closure of complex numbers
	3.7 C is closed under the arithmetic operations
		3.7.1 Sympy examples of arithmetic operations on complex numbers
		3.7.2 Numpy examples of arithmetic operations on complex numbers
	3.8 Examples of nonlinear operations
		3.8.1 The quadratic formula
		3.8.2 Roots of a general polynomial
	3.9 C is also algebraically closed
	3.10 Complex number under nonlinear operations
	3.11 Complex number from transcendental operations
		3.11.1 Exponential operations
		3.11.2 Trigonometric functions
	3.12 C is closed under nonlinear and transcendental operations
	3.13 Some applications of the imaginary part
		3.13.1 Mathematical workaround
		3.13.2 Imaginary unit: A glue
		3.13.3 Singularity bypass
	3.14 Vectors in n-dimensional complex space Cn
	3.15 Remarks
	References
4. Elementary Functions
	4.1 Definition of functions
	4.2 A simple example: A square function
	4.3 Distribution of functions
	4.4 Roots of a function
	4.5 Limits of a function
	4.6 Continuity of a function
	4.7 Reciprocal function
	4.8 Inverse function
	4.9 Linear functions: The most widely used functions
		4.9.1 General form
		4.9.2 Some applications
		4.9.3 Root of the linear function
		4.9.4 Reciprocal of linear function
		4.9.5 Inverse of linear function
	4.10 Monomial functions
		4.10.1 Definition
		4.10.2 Reciprocal of monomial functions
		4.10.3 Inverse of monomial functions
	4.11 Complex-valued functions, closure in C
		4.11.1 A typical example
		4.11.2 Closure of the square function in C
		4.11.3 Functions in general
	4.12 Trigonometric functions
		4.12.1 Definition
		4.12.2 Example: Sine function as the vibration modes of strings
	4.13 Closure of trigonometric functions in C
		4.13.1 Trigonometric identities
		4.13.2 Reciprocal of trigonometric functions
		4.13.3 Inverse of trigonometric functions
		4.13.4 The sinc(x) function
		4.13.5 The sin(1/x) function
		4.13.6 Euler’s formula for trigonometric functions
	4.14 Polynomial functions
		4.14.1 General form
		4.14.2 Major properties
		4.14.3 Numpy codes and examples for polynomials
		4.14.4 Quadratic functions
		4.14.5 Inverse of a polynomial function
		4.14.6 Python code to generate the inverse function of a polynomial
	4.15 Rational functions
		4.15.1 Definition
		4.15.2 Properties
		4.15.3 Example of a converging rational function
		4.15.4 Example of a diverging rational function
		4.15.5 Rational activation function
		4.15.6 Major properties of rational functions
		4.15.7 Example of rational activation function
	4.16 Exponential functions
		4.16.1 Definition
		4.16.2 Python examples
		4.16.3 Major properties of exponential functions
		4.16.4 Sympy examples
		4.16.5 Numpy examples
		4.16.6 Useful functions written in ex
		4.16.7 Example: Decaying oscillatory functions
	4.17 Logarithm functions
		4.17.1 Definition
		4.17.2 Change of bases
		4.17.3 Mutual inverse with the exponential function
		4.17.4 Major properties
		4.17.5 Python examples of functions
		4.17.6 Logarithmic scaling of functions
		4.17.7 Logarithmic scaling for exponential functions
		4.17.8 In complex domain
	4.18 The xx function
	4.19 Composite functions
		4.19.1 Definition and change of domains
		4.19.2 Composition in order
	4.20 Extended topics on functions
		4.20.1 Scalar functions in multi-dimensions
		4.20.2 Vector functions
	4.21 Remarks
5. Basis Functions
	5.1 Vector space of functions
	5.2 Conditions for being basis functions
	5.3 Monomial basis functions
		5.3.1 Monomials
		5.3.2 Proof of monomials as basis functions
		5.3.3 Inverse monomials
		5.3.4 Python examples: Generation of monomial basis
	5.4 Lagrange polynomials
		5.4.1 Definition and formulation
		5.4.2 Properties of Lagrange interpolation
		5.4.3 Proof Lagrange interpolators as basis functions
		5.4.4 Python codes for generating Lagrange interpolators
		5.4.5 Lagrange basis functions in natural coordinate system
		5.4.6 Python codes for the distribution of Lagrange interpolators
	5.5 Chebyshev polynomials
		5.5.1 Formulation and recursive formula
		5.5.2 Some essential properties of Chebyshev polynomials of the first kind
		5.5.3 Chebyshev polynomials of the second kind
		5.5.4 Python codes to generate Chebyshev polynomials
		5.5.5 Roots of Chebyshev polynomials
		5.5.6 Distribution of Chebyshev polynomials
		5.5.7 Conversion between Chebyshev and primitive polynomials
	5.6 Legendre polynomials
		5.6.1 Definition, Bonnet’s recursive formula
		5.6.2 Some essential properties
		5.6.3 Python codes to generate Legendre polynomials
		5.6.4 Roots of the Legendre polynomials
		5.6.5 Distribution of the Legendre polynomials
		5.6.6 Conversion between Legendre and primitive polynomials
	5.7 Laguerre polynomials
		5.7.1 Definition, the recursive formula
		5.7.2 Some essential properties
		5.7.3 Python codes to generate Laguerre polynomials
		5.7.4 Roots of the Laguerre polynomials
		5.7.5 Distribution of the Laguerre polynomials
		5.7.6 Conversion between Laguerre and primitive polynomials
	5.8 Hermite polynomials
		5.8.1 Definition, the recursive formula
		5.8.2 Some essential properties of Hn(x)
		5.8.3 Probabilist’s Hermite polynomials
		5.8.4 Python codes to generate Hermite polynomials
		5.8.5 Roots of the Hermite polynomials
		5.8.6 Distribution of the Hermite polynomials
		5.8.7 Conversion between Hermite and primitive polynomials
	5.9 Shape function: Node-based basis functions
		5.9.1 Formulation in the physical coordinate and properties
		5.9.2 Proof of the linear reproducibility
		5.9.3 Formulation in the natural coordinate
		5.9.4 Python code
	5.10 Remarks
	References
6. Function Approximation
	6.1 Use of monomial basis functions
		6.1.1 Formulation
		6.1.2 Python examples: Approximating a polynomial
		6.1.3 Python examples: Approximating an arbitrary function
	6.2 Use of Lagrange polynomials
		6.2.1 Formulation
		6.2.2 Overfitting phenomenon
		6.2.3 Python example: Approximating a polynomial
		6.2.4 Python example: Approximating an arbitrary function
		6.2.5 Sympy built-in Lagrange interpolate
	6.3 Use of Chebyshev polynomials
		6.3.1 Formulation
		6.3.2 Python example: Approximating a polynomial
		6.3.3 Python example: Approximating an arbitrary function
	6.4 Use of Legendre polynomials
		6.4.1 Formulation
		6.4.2 Python example: Approximating a polynomial
		6.4.3 Python example: Approximating an arbitrary function
	6.5 Use of Laguerre polynomials
		6.5.1 Formulation
		6.5.2 Python example: Approximating a polynomial
		6.5.3 Python example: Approximating an arbitrary function
	6.6 Use of Hermite polynomials
		6.6.1 Python example: Approximating a polynomial
		6.6.2 Python example: Approximating an arbitrary function
	6.7 Use of shape functions
		6.7.1 Formulation and properties
		6.7.2 Python code for approximating a function
	6.8 Remarks
	References
Index