In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function is defined for all complex numbers except non-positive integers, and for every positive integer , The gamma function can be defined via a ...

4 天之前 · The gamma function can be defined as a definite integral for (Euler's integral form) or. The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function . Plots of the real and imaginary parts of in the complex plane are illustrated above.

Many functions have been discovered with those properties. They each have good and bad points. The one most liked is called the Gamma Function (Γ is the Greek capital letter Gamma): It is a definite integral with limits from 0 to infinity. It matches the factorial function for whole numbers (but sadly we must subtract 1):

Definition: Gamma Function The Gamma function is defined by the integral formula

2024年12月10日 · gamma function, generalization of the factorial function to nonintegral values, introduced by the Swiss mathematician Leonhard Euler in the 18th century. For a positive …

The gamma function, denoted by \Gamma (s) Γ(s), is defined by the formula \Gamma (s)=\int_0^ {\infty} t^ {s-1} e^ {-t}\, dt, Γ(s) = ∫ 0∞ ts−1e−tdt, which is defined for all complex numbers except the nonpositive integers. It is frequently used in identities and proofs in analytic contexts.

The Gamma function is a generalization of the factorial function to non-integer numbers. It is often used in probability and statistics, as it shows up in the normalizing constants of important probability distributions such as the Chi-square and the Gamma.

The gamma function is used in the mathematical and applied sciences almost as often as the well-known factorial symbol . It was introduced by the famous mathematician L. Euler (1729) as a natural extension of the factorial operation from positive integers to real and even complex values of the argument .

The gamma functions are used throughout mathematics, the exact sciences, and engineering. In particular, the incomplete gamma function is used in solid state physics and statistics, and the logarithm of the gamma function is used in discrete mathematics, number theory, …

The Gamma function is a generalization of the factorial function. In fact, we have