불완전 감마함수 (Incomplete Gamma Function)
정의
$$ \Gamma(a,x) = \int_{x}^{\infty} t^{a-1} a^{-t} \ dt $$
만약, $a$가 정수라면
$$ \begin{displaymath}\begin{split} \Gamma(a,x) &= (a-1)! \cdot e^{-x} \sum_{k=0}^{a-1} \frac{x^{k}}{k!} \\ &= (a-1)! \cdot e^{-x} \cdot e_{a-1}(x) \end{split}\end{displaymath} $$ <plot> set size 1 set xrange [0:8] set yrange [-1:8] set zeroaxis set title "Incomplete Gamma Function" f(x,a) = (gamma(a)-igamma(a,x)) plot f(x,1.0) title "n=1", \ f(x,2.0) title "n=2", \ f(x,3.0) title "n=3", \ f(x,4.0) title "n=4" </plot><plot> set size 1 set xrange [0:8] set yrange [0:8] set zrange [-1:8] set zeroaxis f(x,a) = (gamma(a)-igamma(a,x)) splot f(x,y) title "" </plot> ---- $$ \begin{displaymath}\begin{split} \gamma(a,x) &= \int_{0}^{x} t^{a-1} a^{-t} \ dt \\ &= a^{-1} \cdot x^{a} \cdot _{1}F_{1} (a;1+a;-x) \end{split}\end{displaymath} $$ 만약, $a$가 정수라면
$$ \begin{displaymath}\begin{split} \gamma(a,x) &= (a-1)! \cdot \left( 1 - e^{-x} \sum_{k=0}^{a-1} \frac{x^{k}}{k!} \right) \\ &= (a-1)! \cdot \left[ 1 - e^{-x} \cdot e_{a-1}(x) \right] \end{split}\end{displaymath} $$
<plot>
set size 1
set xrange [0:8]
set yrange [-1:2]
set zeroaxis
set title "Incomplete Gamma Function"
f(x,a) = (igamma(a,x))
plot f(x,1.0) title "n=1", \
f(x,2.0) title "n=2", \
f(x,3.0) title "n=3", \
f(x,4.0) title "n=4"
</plot><plot>
set size 1
set xrange [0:8]
set yrange [0:8]
set zrange [-1:8]
set zeroaxis
f(x,a) = (igamma(a,x))
splot f(x,y) title ""
</plot>
단, $e_{n}(x)$는 [[지수합 함수]], $_{1}F_{1}=(a;b;z)$는 ????함수이다. 