利用留数定理对一类广义积分的推广

THE GENERALIZATION OF A KIND OF GENERALIZED INTEGRAL BY THE RESIDUE THEOREM

在数学物理问题中常常遇到一类广义积分I±=∫^(∞)_(0)x^(α-1)/1±x^(n)dx或I'±=∫^(∞)_(-∞)x^(α-1)1±xA^(n)dx,其中正整数n和实数α满足n≥1和0<α<n。现有数学物理方法教科书中,一般将n和α取一些特殊值而不具有普遍意义,并且所选积分回路内含有多个奇点,计算较为复杂。我们从整个大圆弧中截取了弧度为2π/n的大圆弧构成的最小扇形积分回路,既推广了参数范围,推导过程又简单。根据留数定理,结果简洁:I+=π/ncscαπ/n,I-=π/ncotαπ/n,和I'±是I±的线性组合。我们综合介绍了此广义积分的几种解法,分析了由高等数学知识求解此类实变函数积分的限制性,并讨论了传统方法的繁琐性。

In the mathematical physics problem,for calculating a kind of generalized integral I±=∫^(∞)_(0)x^(α-1)/1±x^(n)dxorI'±=∫^(∞)_(-∞)x^(α-1)1±xA^(n)dx,where n is a positive integer andαis a real number with n≥1 and 0<α<n,the recent textbooks have taken some special values of n andα,and selected a half complex-plane as a loop that includes many singularities.The calculations are very cumbersome and not universally applicable.We choose the minimum sector loop with a large arc and 2π/n a radian,extend the regime of parameters,and simplify the calculations.Based on the residue theorem,our brief results are I+=π-ncscαπ-n,I-=π-ncotαπ-n,and I'±are expressed as the linear combination of I±.We review several methods to solve this integration,analyze the restrictive nature of this kind of integration with real variables in advanced mathematics textbooks,and also discuss the complexity and cumbersome of the integration in the traditional methods.