摘自https://en.wikipedia.org/wiki/Incomplete_gamma_function

文字定义：解释为什么称为‘不完全’？

Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.

0、ordinary伽马函数

1、上不完全伽马函数

2、下不完全伽马函数

In both cases s is a complex parameter, such that the real part of s is positive.

3、重要关系式

By integration by parts （分部积分法）we find the recurrence relations（递推关系）

power series expansion幂级数展开：

与融合超几何函数的关系：

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x：

4、特殊值

指数积分：

广义指数积分：

误差函数：

互补误差函数：

5、Asymptotic behavior 渐进结果

6、Regularized Gamma functions and Poisson random variables

7、导数

                                      、

the second derivative by

internal closure properties：

where  is the permutation （排列）defined by the Pochhammer symbol（阶乘幂）：