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\mondai
{g140826 $B?eLn(B $BMNJe(B}
{01/12/25}
{g140826@mail.ecc.u-tokyo.ac.jp}
{9-3}
{
$B%Y!<%?4X?t(B$B(n,m)$,$B%,%s%^4X?t(B$\Gamma (z)$$B$O!"(B
\[
B(n,m) = \int_0^1 {t^{n - 1}\left( {1 - t} \right)^{m - 1}dt} 
\]
\[
\Gamma (z) = \int_0^\infty {e^{ - t}t^{z - 1}dt} 
\]
$B$GI=$5$l$k!#0J2<$N8x<0$r<($;!#(B
$$(a)\qquad  \Gamma (\ell + 1) = \ell!\qquad$$
$$(b)\quad B(n,m) = \frac{\Gamma (n)\Gamma (m)}{\Gamma (n + m)}$$
}
{
(a)$B$^$:!"ItJ,@QJ,$rMQ$$$F!"(B
\begin{eqnarray}
\Gamma (\ell + 1) &=& \int_0^\infty {e^{ - t}t^\ell dt}\nonumber\\
& = &\left[ { - e^{ - t}t^\ell} \right]_0^\infty + \int_0^\infty {e^{ - t}\ell t^{\ell - 1}dt}\nonumber\\
& = &\mathop {\lim }\limits_{M \to \infty } 
 \left( { - \frac{M^\ell}{e^M}} \right) + \ell\int_0^\infty {e^{ - t}t^{\ell - 1}dt}\nonumber\\
& = &0 + \ell\Gamma (\ell)\nonumber\\
& = &\ell\Gamma (\ell)
\end{eqnarray}
$B$rF@$k!#$^$?!"(B
\begin{eqnarray}
\Gamma (1) &=& \int_0^\infty {e^{ - t}dt}\nonumber\\
& = &\left[ { - e^{ - t}} \right]_0^\infty\nonumber\\
& = &0 + 1\nonumber\\
& = &1
\end{eqnarray}
$B$G$"$j!"(B(9-3-1)$B$r7+$jJV$7MQ$$$k$H!"(B
\begin{eqnarray}
\Gamma (\ell + 1) &=& \ell\Gamma (\ell)\nonumber\\
& = &\ell(\ell - 1)\Gamma (\ell - 1)\nonumber\\
& = &\ell!\Gamma (1)\nonumber\\
& = &\ell!
\end{eqnarray}
$B$H$J$k!#(B\\

(b)$B%,%s%^4X?t$N@Q$O!"(B
\begin{eqnarray}
& &\Gamma (n)\Gamma (m) \nonumber\\
& = & \int_0^\infty {e^{ - t}t^{n - 1}dt} \int_0^\infty {e^{ - s}s^{m - 1}ds}\nonumber\\
& = &\int_0^\infty {\int_0^\infty {t^{n - 1}s^{m - 1}e^{ - (t + s)}} dtds}\nonumber\\
& = &\int_0^\infty {\int_0^\infty {\left( {\frac{t}{t + s}} 
\right)^{n - 1}\left( {\frac{s}{t + s}} \right)^{m - 1}\left( {t + s} 
\right)^{n + m - 2}e^{ - (t + s)}} dtds} \nonumber\\
\end{eqnarray}
$B$G$"$k!#$3$3$G!"(B
\begin{eqnarray}
u = \frac{t}{t + s}\nonumber\\
v = t + s
\end{eqnarray}
$B$HJQ?tJQ49$9$k$H!"(B
\begin{eqnarray}
t = u(t + s) = uv
\end{eqnarray}
\begin{eqnarray}
s = v - t = v - uv = v(1 - u)
\end{eqnarray}
$B$G$"$j!"JQ49$N%d%3%S%"%s$O!"(B
\begin{eqnarray}
\begin{displaystyle}
\left| {{\begin{array}{cc}
 \displaystyle{\frac{\partial t}{\partial u}}  & \displaystyle{\frac{\partial t}{\partial v}}  \\
 \displaystyle{\frac{\partial s}{\partial u}}  & \displaystyle{\frac{\partial s}{\partial v}}  \\
\end{array} }} \right| = \left| {{\begin{array}{cc}
 v  & u  \\
 { - v}  & {1 - u}  \\
\end{array} }} \right| = v(1 - u) - ( - uv) = v
\end{displaystyle}
\end{eqnarray}
$B$H$J$k!#@QJ,HO0O$O!"(B
\begin{eqnarray}
0 \le u \le 1$B!"(B
0 \le v < \infty 
\end{eqnarray}
$B$G$"$k$+$i!"7k6I!"(B
\begin{eqnarray}
\Gamma (n)\Gamma (m) &=& \int_0^1 {\int_0^\infty {u^{n - 1}
\left( {1 - u} \right)^{m - 1}v^{n + m - 2}e^{ - v}v} dudv}\nonumber\\
& = & \int_0^1 {u^{n - 1}\left( {1 - u} \right)^{m - 1}du} 
\int_0^\infty {v^{n + m - 1}e^{ - v}dv}\nonumber\\
& = &B(n,m)\Gamma (n + m)
\end{eqnarray}
$B$,F@$i$l$k!#$h$C$F!"(B
\begin{eqnarray}
B(n,m) = \frac{\Gamma (n)\Gamma (m)}{\Gamma (n + m)}
\end{eqnarray}
}
