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Approximation for the gamma function via the tri-gamma function.

Xu You1, Xiaocui Li2

  • 1Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 102617 P.R. China.

Journal of Inequalities and Applications
|December 22, 2017
PubMed
Summary
This summary is machine-generated.

Researchers developed a new, fast approximation for the gamma function using the tri-gamma function. This method offers superior accuracy and efficiency compared to existing asymptotic series, supported by numerical evidence.

Keywords:
approximationgamma functioninequalitiesmultiple-correction method

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Area of Science:

  • Mathematical Analysis
  • Special Functions

Background:

  • The gamma function is a fundamental concept in mathematical analysis, crucial for various scientific fields.
  • Existing approximations, such as asymptotic series, can be computationally intensive.

Purpose of the Study:

  • To introduce a novel, sharp approximation for the gamma function.
  • To enhance computational efficiency in calculations involving the gamma function.
  • To provide theoretical underpinnings and numerical validation for the new approximation.

Main Methods:

  • Derivation of a new approximation formula for the gamma function.
  • Utilizing the tri-gamma function in the approximation development.
  • Establishing inequalities to bound the approximation error.
  • Performing numerical computations to compare performance.

Main Results:

  • A new sharp approximation for the gamma function has been successfully derived.
  • The proposed approximation demonstrates significantly faster computation speeds than existing asymptotic series.
  • Inequalities related to the approximation have been established, providing error bounds.
  • Numerical results confirm the superiority and accuracy of the new approximation.

Conclusions:

  • The new tri-gamma function-based approximation offers a more efficient and accurate alternative for gamma function calculations.
  • This advancement has potential implications for fields relying on precise gamma function evaluations.
  • The established inequalities provide a rigorous framework for understanding the approximation's behavior.