幾個定積分

<正> 在多個複變數函数論Ⅱ1]一文的§6中(參考本文附錄),我們曾經用間接的方法算出了以下定積分的数值:若n(≥3),m,l是正整數,則

SOME DEFINITE INTEGRALS

It was proved indirectlyi that for n(≥3), m, n being integers, then where z is a vector with n complex components and zz' P_m~(ν)(t) is the ultraspherical polynomial of the m-th degree. The aim of the paper is to prove (1) directly.Firstly, we reduce (1) into a simple integral which is a special case of the following result: Let s be a complex number with real part > m + 2λ, λ, be a real number >-1/2. Then, we haveThe proof depends essentially on the following facts: Substituting P_m~(λ)(x) by its expansion and integrating term by term, we obtain a generalized hypergeometric series. Fortunately the series is of Saalschutz's type, it can be expressed in terms of gamma function.