(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*}