第1種・第2種不完全ガンマ関数の基本性質

by nomura · 2021年3月2日

第1種・第2種不完全ガンマ関数の基本性質

(1)

\[ \Gamma\left(a,0\right)=\Gamma\left(a\right) \]

(2)

\[ \lim_{x\rightarrow\infty}\gamma\left(a,x\right)=\Gamma\left(a\right) \]

(3)

\[ \Gamma\left(1,x\right)=e^{-x} \]

(4)

\[ \gamma\left(1,x\right)=1-e^{-x} \]

(5)

\[ \Gamma\left(\frac{1}{2},x\right)=\sqrt{\pi}erfc\left(\sqrt{x}\right) \]

(6)

\[ \gamma\left(\frac{1}{2},x\right)=\sqrt{\pi}\erf\left(\sqrt{x}\right) \]

(7)

$x>0$のとき、
\[ \Gamma\left(0,x\right)=-\Ei\left(-x\right) \]

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$\gamma\left(a,x\right)$は第1種不完全ガンマ関数、$\Gamma\left(a,x\right)$は第2種不完全ガンマ関数
$\erf\left(x\right)$は誤差関数、$erfc\left(x\right)$は相補誤差関数、$\Ei\left(x\right)$は指数積分

(1)

\begin{align*} \Gamma\left(a,0\right) & =\int_{0}^{\infty}t^{a-1}e^{-t}dt\\ & =\Gamma\left(a\right) \end{align*}

(2)

\begin{align*} \lim_{x\rightarrow\infty}\gamma\left(a,x\right) & =\lim_{x\rightarrow\infty}\int_{0}^{x}t^{a-1}e^{-t}dt\\ & =\int_{0}^{\infty}t^{a-1}e^{-t}dt\\ & =\Gamma\left(a\right) \end{align*}

(3)

\begin{align*} \Gamma\left(1,x\right) & =\int_{x}^{\infty}e^{-t}dt\\ & =\left[-e^{-t}\right]_{x}^{\infty}\\ & =e^{-x} \end{align*}

(4)

\begin{align*} \gamma\left(1,x\right) & =\int_{0}^{x}e^{-t}dt\\ & =\left[-e^{-t}\right]_{0}^{x}\\ & =1-e^{-x} \end{align*}

(5)

\begin{align*} \Gamma\left(\frac{1}{2},x\right) & =\int_{x}^{\infty}t^{\frac{1}{2}-1}e^{-t}dt\\ & =\int_{x}^{\infty}t^{-\frac{1}{2}}e^{-t}dt\\ & =2\int_{x}^{\infty}e^{-s^{2}}ds\qquad,\qquad t=s^{2}\\ & =2\int_{\sqrt{x}}^{\infty}e^{-s^{2}}ds\\ & =\sqrt{\pi}erfc\left(\sqrt{x}\right) \end{align*}

(6)

\begin{align*} \gamma\left(\frac{1}{2},x\right) & =\int_{0}^{x}t^{\frac{1}{2}-1}e^{-t}dt\\ & =\int_{0}^{x}t^{-\frac{1}{2}}e^{-t}dt\\ & =2\int_{0}^{\sqrt{x}}e^{-s^{2}}ds\qquad,\qquad t=s^{2}\\ & =\sqrt{\pi}\erf\left(\sqrt{x}\right) \end{align*}

(7)

$x>0$のとき、
\begin{align*} \Gamma\left(0,x\right) & =\int_{x}^{\infty}t^{-1}e^{-t}dt\\ & =-\Ei\left(-x\right) \end{align*}