Package org.drip.specialfunction.property

Class GammaInequalityLemma

java.lang.Object
org.drip.specialfunction.property.GammaInequalityLemma
public class GammaInequalityLemma
extends java.lang.Object
GammaInequalityLemma contains the Verifiable Inequality Lemmas of the Gamma Function. The References are:

  • Blagouchine, I. V. (2014): Re-discovery of Malmsten's Integrals, their Evaluation by Contour Integration Methods, and some Related Results Ramanujan Journal 35 (1) 21-110
  • Borwein, J. M., and R. M. Corless (2017): Gamma Function and the Factorial in the Monthly https://arxiv.org/abs/1703.05349 arXiv
  • Davis, P. J. (1959): Leonhard Euler's Integral: A Historical Profile of the Gamma Function American Mathematical Monthly 66 (10) 849-869
  • Whitaker, E. T., and G. N. Watson (1996): A Course on Modern Analysis Cambridge University Press New York
  • Wikipedia (2019): Gamma Function https://en.wikipedia.org/wiki/Gamma_function
It provides the following functionality:
  • Construct the Asymptotic Upper Approximate
  • Generate the Exponentially Convex Inequality Verifier
  • Generate the Spaced Point Convex Inequality Verifier
  • Generate the Logarithmically Convex Inequality Verifier
  • Generate the Gautschi Left Inequality Verifier
  • Generate the Gautschi Right Inequality Verifier
  • Generate the Jensen Multi-Point Interpolant Convexity Verification

Module Computational Core Module
Library Function Analysis Library
Project Special Function Implementation and Analysis
Package Special Function Property Lemma Verifiers
Author:
Lakshmi Krishnamurthy

  • Constructor Details

    • GammaInequalityLemma

      public GammaInequalityLemma()

  • Method Details

    • AsymptoticUpperApproximate

      public static final R1ToR1Property AsymptoticUpperApproximate​(double alpha)
      Construct the Asymptotic Upper Approximate
      Parameters:
      alpha - Alpha
      Returns:
      The Asymptotic Upper Approximate

    • ExponentiallyConvex

      public static final R1ToR1Property ExponentiallyConvex​(double z1, double z2)
      Generate the Exponentially Convex Inequality Verifier
      Parameters:
      z1 - z1
      z2 - z2
      Returns:
      The Exponentially Convex Inequality Verifier

    • SpacedPointConvex

      public static final R1ToR1Property SpacedPointConvex​(double y)
      Generate the Spaced Point Convex Inequality Verifier
      Parameters:
      y - y
      Returns:
      The Spaced Point Convex Inequality Verifier

    • LogarithmicConvex

      public static final R1ToR1Property LogarithmicConvex()
      Generate the Logarithmically Convex Inequality Verifier
      Returns:
      The Logarithmically Convex Inequality Verifier

    • GautschiLeft

      public static final R1ToR1Property GautschiLeft​(double s)
      Generate the Gautschi Left Inequality Verifier
      Parameters:
      s - s
      Returns:
      The Gautschi Left Inequality Verifier

    • GautschiRight

      public static final R1ToR1Property GautschiRight​(double s)
      Generate the Gautschi Right Inequality Verifier
      Parameters:
      s - s
      Returns:
      The Gautschi Right Inequality Verifier

    • JensenMultiPointInterpolant

      public static final R1PropertyVerification JensenMultiPointInterpolant​(Array2D multiPoint2D)
      Generate the Jensen Multi-Point Interpolant Convexity Verification
      Parameters:
      multiPoint2D - Multi-Point 2D
      Returns:
      Jensen Multi-Point Interpolant Convexity Verification