ウォリス積分
! = 1 2 n + 1 ⋅ 4 n 2 n C n {\displaystyle I_{2n+1}={\frac {2n}{2n+1}}\cdot {\frac {2n-2}{2n-1}}\cdots {\frac {2}{3}}\cdot 1={\frac {(2n)!!}{(2n+1)!!}}={\frac {1}{2n+1}}\cdot {\frac {4^{n}}{{}_{2n}{\rm {C}}_{n}}}}
I 2 n = 2 n − 1 2 n ⋅ 2 n − 3 2 n − 2 ⋯ 1 2 ⋅ π 2 = ( 2 n − 1 ) ! ! ( 2 n ) ! ! ⋅ π 2 = 2 n C n 4 n ⋅ π 2 {\displaystyle I_{2n}={\frac {2n-1}{2n}}\cdot {\frac {2n-3}{2n-2}}\cdots {\frac {1}{2}}\cdot {\frac {\pi }{2}}={\frac {(2n-1)!!}{(2n)!!}}\cdot {\frac {\pi }{2}}={\frac {{}_{2n}{\rm {C}}_{n}}{4^{n}}}\cdot {\frac {\pi }{2}}}