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The well-known multiple integralwhere R<sub>n</sub> is the region defined by x<sub>1</sub> ≥ 0, x<sub>2</sub> ≥ 0, …., x<sub>n</sub> ≥ 0, x<sub>1</sub> + x<sub>2</sub> + …. + x<sub>n</sub> ≤ 1, and where a<sub>0</sub>, a<sub>1</sub>, …, a<sub>n</sub> are positive constants, can be evaluated either in the classical way using the Dirichlet transformation or by the use of the Laplace transform. I. J. Good has considered a more general integral and has proved the following result by induction:—If f<sub>1</sub>(t), f<sub>2</sub>(t), …, f<sub>n</sub>(t) are Lebesgue measurable for 0 ≤ t ≤ 1, m<sub>1</sub>, m<sub>2</sub>, …., m<sub>n</sub>, m<sub>n+1</sub> (= 0) are real numbers, M<sub>r</sub> = m<sub>1</sub> + m<sub>2</sub> + … + m<sub>r</sub>, x<sub>1</sub>, x<sub>2</sub>, …, x<sub>n</sub> are non-negative variables and X<sub>r</sub> = x<sub>1</sub> + x<sub>2</sub> + … + x<sup>r</sup>, thenIt does not seem to be possible to establish this relation by employing the Laplace transform, but we show below that it can be obtained using the Mellin transform. |
