不完全ガンマ関数
∂ Γ ( a , x ) ∂ x = − x a − 1 e x {\displaystyle {\frac {\partial \Gamma (a,x)}{\partial x}}=-{\frac {x^{a-1}}{e^{x}}}}
MeijerのG関数から:
T ( m , a , x ) = G m − 1 , m m , 0 ( 0 , 0 , ... , 0 a − 1 , − 1 , ... , − 1 | x ) {\displaystyle T(m,a,x)=G_{m-1,\,m}^{\,m,\,0}\!\left(\left.{\begin{matrix}0,0,\dots ,0\\a-1,-1,\dots ,-1\end{matrix}}\;\right|\,x\right)}
T ( m , a , z ) = − ( − 1 ) m − 1 ( m − 2 ) ! d m − 2 d t m − 2 [ Γ ( a − t ) z t − 1 ] | t = 0 + ∑ n = 0 ∞ ( − 1 ) n z a − 1 + n n !
( − a − n ) m − 1 {\displaystyle T(m,a,z)=-{\frac {(-1)^{m-1}}{(m-2)!}}{\frac {{\rm {d}}^{m-2}}{{\rm {d}}t^{m-2}}}\left[\Gamma (a-t)z^{t-1}\right]{\Big |}_{t=0}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{a-1+n}}{n!(-a-n)^{m-1}}}} 時 | z | < 1 {\displaystyle |z|<1}
∂ Γ ( a , x ) ∂ a = ln  x Γ ( a , x ) + x T ( 3 , a , x ) {\displaystyle {\frac {\partial \Gamma (a,x)}{\partial a}}=\ln x\Gamma (a,x)+x\,T(3,a,x)}
∂ 2 Γ ( a , x ) ∂ a 2 = ln 2  x Γ ( a , x ) + 2 x [ ln  x T ( 3 , a , x ) + T ( 4 , a , x ) ] {\displaystyle {\frac {\partial ^{2}\Gamma (a,x)}{\partial a^{2}}}=\ln ^{2}x\Gamma (a,x)+2x[\ln x\,T(3,a,x)+T(4,a,x)]}
γ ( a , x ) = ∫ 0 x t a − 1 e − t d t {\displaystyle \gamma (a,x)=\int _{0}^{x}t^{a-1}\,e^{-t}\,dt\,\!}