Gamma function _ gamma

Meaning

Gamma Function (Gamma Function), as an extension of factorial, is a family of objective functions defined in the plural range. It is usually written as Gamma (x ).

When the variable of a function is a positive integer, the value of the function is the factorial of the previous integer, or gamma (n + 1) = n !.

Formula

Gamma function expression: gamma (x) = ∫ e ^ (-t) * t ^ (x-1) dt (the lower limit of points is 0, the upper limit is + ∞)

By using the integration by parts, we can obtain the gamma (x) = (x-1) * gamma (x-1), and it is easy to calculate the gamma (1) = 1,

From this we can see that the range of positive integers is gamma (n + 1) = n!

An important distribution in the study of probability is calledGamma Distribution: F (x) = λ e ^ (-λ x) (λ x) ^ (x-1)/gamma (x) x> = 0 = 0x<0

Function expression: Right

Nature

Gamma (x + 1) = x gamma (x), gamma (0) = 1, gamma (1/2) = √ π, for positive integer n, there are gamma (n + 1) = n !, For x> 0, gamma (x) = π/sin (π x) is a strictly convex function.

The Gamma function is a sub-family function. On the repeat plane, except for the zero and negative integer points, it resolves all the points, and the number of records of the Gamma function at-k is (-1) ^ k/k!

Stealin progressive

Stirling's approximation: ln gamma (x) = (x-1/2) ln (x)-x + ln (2 π) /2 + Σ B _ {2n}/(2n (2n-1) x ^ {2n-1}). Here, n in the sum following ranges from 1 to infinity. Among them, B _ {2n} is the number of Bay effort. The first few Bay effort is B _2 = 1/6, B _4 =-1/30, B _6 = 1/42, B _8 =-1/30, B _10 = 5/66, etc. In general, we usually remove all the following Σ, and call it the styline formula.

Digamma Function

The natural logarithm differentiation of the Gamma function is called the Digamma function, which is recorded as follows: (x) = d (ln gamma (x)/dx = gamma '(x)/gamma (x ).

The Digamma function is related to the harmonic series, where Gini (n + 1) = H_n (x)-Gamma = 1 + 1/2 +... + 1/n-Gamma, where Gamma = lim _ {n-> infty} (1 + 1/2 +... + 1/n-ln (n) is Euler's constant.

For any x, returns (x + 1) = returns (x) + 1/x.

In the range of plural numbers, the Digamma function can be written as cosine (x + 1) =-Gamma + Σ x/(n + x )).

The Taylor expansion of the Digamma function is round (x + 1) =-gamma-Σ ε (n + 1) (-x) ^ n, where the function ε (x) this is an important function about the Cartesian conjecture.

Similar to the Gamma function, the Digamma function can be progressive: round (x) = ln (x)-1/(2x) -Σ B _ {2n}/(2n * x ^ {2n })

From: Baidu encyclopedia Co., http://baike.baidu.com/view/4898730.htm.