scipy 中不完整的 Gamma 函数 [英] Incomplete Gamma function in scipy

问题描述

我想计算 wolfram alpha 调用的不完全伽马函数

根据 http://mathworld.wolfram.com/IncompleteGammaFunction.html，这是上不完全伽马函数，而 Scipy 输出的是 WolframAlpha 称为下不完全伽马函数的缩放版本.通过使用等式 10 中的恒等式，可以看到在 a>0 的情况下，您可以使用以下内容:

from scipy.special import gammainc来自 scipy.special 导入伽马伽马(0.01)*(1 - 伽马inc(0.01,0.1))

与 WolframAlpha 一致返回 1.8032413569025461.

简而言之，WolframAlpha 中的 Gamma[a,x] 对应于 Scipy 中的 gamma(a)*(1-gammainc(a,x))，前提是a>0.

I would like to compute what wolfram alpha calls the incomplete gamma function (see here):

`gamma[0, 0.1]`

The wolfram alpha output is 1.822. The only thing scipy gives me that resembles this is scipy.special.gammainc, but it has a different definition than how wolfram alpha defines their incomplete gamma function.

Not surprisingly

import scipy
scipy.special.gammainc(0, 0.1)

gives me nan. Does scipy support what I'm looking for?

解决方案

According to http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.special.gammainc.html, the first argument must be positive, whereas you have zero; that's why you're getting NaN.

That said, suppose we try to compute Gamma[0.01,0.1] instead. In this case WolframAlpha returns 1.80324:

According to http://mathworld.wolfram.com/IncompleteGammaFunction.html, this is the Upper Incomplete Gamma Function, whereas what Scipy outputs is a scaled version of what WolframAlpha calls the Lower Incomplete Gamma Function. By using the identity in Equation 10, one can see that in cases where a>0, you can use the following:

from scipy.special import gammainc
from scipy.special import gamma
gamma(0.01)*(1 - gammainc(0.01,0.1))

which returns 1.8032413569025461 in agreement with WolframAlpha.

In short, Gamma[a,x] in WolframAlpha corresponds to gamma(a)*(1-gammainc(a,x)) in Scipy, provided that a>0.

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