物理与工程

在积分区间[0,∞]上，运用留数定理方法讨论了被积函数分别为1/(1+xⁿ)和1/(1-xⁿ)的两类无穷积分。由于被积函数1/(1-xⁿ)中包含瑕点x=1,所以其积分路径必须绕开瑕点。采用不同的圆弧积分路径，得到相同的积分结果，即1/(1+xⁿ)的积分为(π/n)·csc(π/n),1/(1-xⁿ)的积分为(π/n)·cot(π/n)。在当前的数学物理方法教科书中，很少涉及这两类积分的讨论，因此这两类无穷积分可作数学物理方法课程中留数定理应用的典型例题。

Within the integral interval[0,∞], two types of infinite integrals with the integrand of 1/(1+xⁿ) and 1/(1-xⁿ) are discussed using method of residue theorem, respectively. There is a defect x=1 in the integrand 1/(1-xⁿ), hence the defect must be excluded from the paths of integration. Over different arc paths of integration, the same results are obtained, namely, the infinite integral with the integrand of 1/(1+xⁿ) is equal to(π/n)·csc(π/n), and the infinite integral with the integrand of 1/(1-xⁿ) is equal to(π/n)·cot(π/n). In current textbooks of mathematical physics methods, the analysis on these two kinds of infinite integrals is still scare. Therefore, these two types of infinite integrals can be used as typical examples of residue theorem in the course of mathematical physics methods.