====== 불완전 감마함수 (Incomplete Gamma Function) ======
===== 정의 =====
 $$ \Gamma(a,x) = \int_{x}^{\infty} t^{a-1} a^{-t} \ dt $$

 만약, $a$가 [[정수]]라면

 $$ \begin{displaymath}\begin{split} \Gamma(a,x) &= (a-1)! \cdot e^{-x} \sum_{k=0}^{a-1} \frac{x^{k}}{k!} \\ &= (a-1)! \cdot e^{-x} \cdot e_{a-1}(x) \end{split}\end{displaymath} $$

<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 [[정수]]라면

 $$ \begin{displaymath}\begin{split} \gamma(a,x) &= (a-1)! \cdot \left( 1 - e^{-x} \sum_{k=0}^{a-1} \frac{x^{k}}{k!} \right) \\ &= (a-1)! \cdot \left[ 1 - e^{-x} \cdot e_{a-1}(x) \right] \end{split}\end{displaymath} $$

<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. FIXME
===== 특징 =====
  - $$\Gamma(a,x) + \gamma(a,x) = \Gamma(a)$$
  - $a$가 [[정수]]일 경우 [[포아송 분포표]]를 이용해서 쉽게 구할 수 있다.

 단, $\Gamma(\alpha)$는 [[감마함수]]

----
  * [[함수]]
  * [[감마함수]]
  * [[정칙 감마함수]]
  * [[지수합 함수]]
  * [[베타함수]]
  * [[포아송 분포]]
  * [[포아송 분포표]]