The integral along curves by residue theorem

Abstract

It has been shown that ∫∞n(𝐥𝐨𝐠 𝒙)𝟐/𝟏+𝒙𝟐 𝒅𝒙 = 𝝅𝟑/𝟖, and 𝐥𝐨𝐠 𝒛 has been chosen to be a branch of the logarithm function in this paper. Meanwhile, logz is holomorphic in the domain: {𝒛: 𝑰𝒎𝒛 ≥ 𝟎 𝒂𝒏𝒅 𝒛 ≠ 𝟎}. Then this study calculates how to express residue of f as 𝟏 + 𝐳𝟐 = (𝐳 − 𝐢) ∙ (𝐳 + 𝐢), and there are two solutions of 𝟏 + 𝐳𝟐 = 𝟎: 𝒛𝟏 = 𝒊 𝒂𝒏𝒅 𝒛𝟐 = −𝒊. The function has only one pole occurs at 𝐳 = 𝐢. This result applies to more calculation of integral along curves.