FunctionAnalysisLibrary.md

Function Analysis Library

Function Analysis Library contains the Libraries implementing the Special Functions and their Analysis.

Documentation

Document	Link
Technical Specification	Latest Previous
User Guide
API	Javadoc

Component Projects

• Special Function => Special Function and their Analysis.
  • { Home | Project }

Coverage

• Gamma Function
  • Introduction and Background
  • Motivation
  • Main Definition
  • Alternate Definitions: Euler’s Definition as an Infinite Product
  • Weierstrass Definition
  • In Terms of Generalized Laguerre Polynomials
  • General Properties
  • Inequalities
  • Stirling’s Formula
  • Residues
  • Minima
  • Integral Representations
  • Fourier Series Expansion
  • Raabe’s Formula
  • Pi Function
  • Relation to Other Functions
  • Particular Values
  • The Log-Gamma Function
  • The Log-Gamma Function Properties
  • Integration Over Log-Gamma
  • Approximations
  • Applications – Integration Problems
  • Calculating Products
  • Analytic Number Theory
  • References
• Stirling's Approximation
  • Introduction and Overview
  • Derivation
  • An Alternative Derivation
  • Speed of Convergence and Error Estimates
  • Stirling’s Formula for the Gamma Function
  • Error Bounds
  • A Convergent Version of the Sterling’s Formula
  • Versions Suitable for Calculators
  • References
• Lanczos Approximation
  • Introduction
  • Coefficients
  • References
• Incomplete Gamma Function
  • Introduction and Overview
  • Definition
  • Properties
  • Continuation to Complex Values
  • Lower Incomplete Gamma FUnction - Holomorphic Extensions
  • Multi-Valuedness
  • Sectors
  • Branches
  • Relationshiop between Branches
  • Behavior near the Branch Point
  • Algebraic Relations
  • Integral Representation
  • Limit for z→±∞ - Real Values
  • Limit for z→±∞ - Complex Values
  • Sector-wise Convergence
  • Lower Incomplete Gamma Function – Overview
  • Upper Incomplete Gamma Function
  • Special Values
  • Asymptotic Behavior
  • Evaluation Formulas
  • Connection with Kummer’s Confluent Hyper-geometric Function
  • Multiplication Theorem
  • Software Implementation
  • Regularized Gamma Functions and Poisson Random Variables
  • Derivatives
  • Indefinite and Definite Integrals
  • References
• Digamma Function
  • Introduction and Overview
  • Relation to the Harmonic Series
  • Integral Representations
  • Infinite Product Representation
  • Series Formula
  • Evaluation of Sums of Rational Functions
  • Taylor Series
  • Newton Series
  • Series with Gregory’s Coefficients, Cauchy Numbers, and Bernoulli Polynomials of the Second Kind
  • Reflection Formula
  • Recurrence Formula and Characterization
  • Some Finite Sums involving the Digamma Function
  • Gauss Digamma Theorem
  • Asymptotic Expansion
  • Inequalities
  • Computation and Approximation
  • Special Values
  • Roots of the Digamma Function
  • Regularization
  • References
• Beta Function
  • Introduction and Overview
  • Properties
  • Relationship between Gamma Function and Beta Function
  • Derivatives
  • Integrals
  • Approximation
  • Incomplete Beta Function
  • Incomplete Beta Function - Properties
  • Multi-variate Beta Function
  • Software Implementation
  • References
• Hypergeometric Function
  • Introduction and Overview
  • The Hypergeometric Series
  • Differentiation Formulas
  • Special Cases
  • The Hypergeometric Differential Equation
  • Solutions at the Singular Points
  • Kummer’s 24 Solutions
  • Q-Form
  • Schwarz Triangle Maps
  • Monodromy Group
  • Integral Formulas – Euler Type
  • Barnes Integral
  • John Transform
  • Gauss’ Contiguous Relation
  • Gauss’ Continued Fraction
  • Transformation Formula
  • Fractional Linear Transformations
  • Quadratic Transformation
  • Higher Order Transformations
  • Values at Special Points
  • Special Values at 𝒛 = 𝟏
  • Kummer’s Theorem
  • Special Values at 𝒛 = 0.5
  • Other Points
  • References
• Bessel Function
  • Introduction and Overview
  • Applications of Bessel Functions
  • Definitions
  • Bessel Functions of the First Kind J_α
  • Bessel’s Integrals
  • Relation to the Hypergeometric Series
  • Relation to the Laguerre Polynomials
  • Bessel Function of the Second Kind Y_α
  • Hankel Functions 〖H_α〗^((1) ) and 〖H_α〗^((2) )
  • Modified Bessel Functions I_α, K_α
  • Spherical Bessel Functions j_n and y_n
  • Generating Function
  • Differential Relations
  • Spherical Hankel Functions
  • Riccati-Bessel Functions - S_n, C_n, ξ_n, ζ_n
  • Asymptotic Forms
  • Full Domain Approximations with Elementary Functions
  • Properties
  • Recurrence Relations
  • Multiplication Theorem
  • Zeros of the Bessel Function – Bourget’s Hypothesis
  • Numerical Approaches
  • References
• Stretched Exponential Function
  • Introduction and Overview
  • Mathematical Properties
  • Mathematical Properties – Distribution Function
  • Fourier Transform
  • Further Applications
  • Use in Probability
  • Modified Stretched Exponential
  • References
• Error Function
  • Introduction and Overview
  • Name
  • Applications
  • Properties
  • Taylor Series
  • Derivative and Integral
  • Burmann Series
  • Inverse Functions
  • Asymptotic Expansion
  • Continued Fraction Expansion
  • Integral of Error Function with Gaussian Density Function
  • Factorial Series
  • Numerical Approximations – Approximation with Elementary Functions
  • Polynomial
  • Table of Values
  • Related Functions – Complementary Error Function
  • Imaginary Error Function
  • Cumulative Distribution Function
  • Generalized Error Functions
  • Iterated Integrals of the Complementary Error Function
  • References

DROP Specifications

• Main => https://lakshmidrip.github.io/DROP/
• Wiki => https://github.com/lakshmiDRIP/DROP/wiki
• GitHub => https://github.com/lakshmiDRIP/DROP
• Repo Layout Taxonomy => https://lakshmidrip.github.io/DROP/Taxonomy.md
• Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
• Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
• Release Versions => https://lakshmidrip.github.io/DROP/version.html
• Community Credits => https://lakshmidrip.github.io/DROP/credits.html
• Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues