【たじゅうせきぶんのかいほう (solution of multiple integral)】
多重積分を累次(繰り返し)積分で解くこと:
∫ D f ( x ) d x = ∫ D 1 ∫ D 2 ( x 1 ) ⋯ ∫ D N ( x 1 , x 2 , ⋯ , x N − 1 ) f ( x 1 , x 2 , ⋯ , x N ) d x N ⋯ d x 2 d x 1 {\displaystyle \displaystyle {\int _{D}f(x){\mbox{d}}x=}\displaystyle {\int _{D_{1}}\int _{D_{2}(x_{1})}\cdots \int _{D_{N}(x_{1},x_{2},\cdots ,x_{N-1})}}\displaystyle {f(x_{1},x_{2},\cdots ,x_{N}){\mbox{d}}x_{N}\cdots {\mbox{d}}x_{2}{\mbox{d}}x_{1}}\,}
は f = f N {\displaystyle f=f_{N}\,} から始まる後向きの再帰(漸化)式
f n − 1 ( x 1 , ⋯ , x n − 1 ) = ∫ D n ( x 1 , ⋯ , x n − 1 ) f n ( x 1 , ⋯ , x n ) d x n , 1 ≤ n ≤ N {\displaystyle \displaystyle {f_{n-1}(x_{1},\cdots ,x_{n-1})=}\displaystyle {\int _{D_{n}(x_{1},\cdots ,x_{n-1})}f_{n}(x_{1},\cdots ,x_{n}){\mbox{d}}x_{n},\ 1\leq n\leq N}\,}
を解くことに他ならない.
から始まる後向きの再帰(漸化)式
