Gamma function, also called Euler's second integral, is a function of factorial function extending on real and complex numbers.

This function has important applications in analysis, probability theory, partial differential equations and combinatorial mathematics. The function closely related to it is beta function, also called Euler integral of the first kind, which can be used to quickly calculate the integral similar to gamma function.

Gamma function has been studied by many mathematicians since its birth, including Gauss, Legendre, Wilstrass, joseph liouville and so on. This function is deeply studied in modern mathematical analysis and ubiquitous in probability theory, and many statistical distributions are related to this function. As a generalization of factorial, gamma function has a conclusion similar to Stirling formula: when the number of x is large, gamma function tends to Stirling formula, so when x is large enough, the value of gamma function can be calculated by Stirling formula.

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Many programming languages or spreadsheet software have gamma functions that provide gamma functions or logarithms, such as EXCEL. Logarithmic gamma function needs to take the natural exponent again to get the gamma function value. For example, in EXCEL, you can use GAMMALN function, and then use EXP[GAMMALN(X)] to get the value of gamma function of any real number.

For example, in EXCEL, EXP [gammaln (4/3)] = 0.89297951156925, but in the program environment without γ function, Taylor series or Stirling formula can also be used for approximation, such as the method proposed by Robert H. Windschitl in 2002, and the decimal eight-digit significant number can be obtained.

Baidu Encyclopedia-Gamma Function