{"title":"用剩余定理求沿曲线的积分","authors":"Yuan Liu","doi":"10.1117/12.2679151","DOIUrl":null,"url":null,"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.","PeriodicalId":301595,"journal":{"name":"Conference on Pure, Applied, and Computational Mathematics","volume":"271 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The integral along curves by residue theorem\",\"authors\":\"Yuan Liu\",\"doi\":\"10.1117/12.2679151\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":301595,\"journal\":{\"name\":\"Conference on Pure, Applied, and Computational Mathematics\",\"volume\":\"271 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Conference on Pure, Applied, and Computational Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1117/12.2679151\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Conference on Pure, Applied, and Computational Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1117/12.2679151","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.
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