用剩余定理求沿曲线的积分

Conference on Pure, Applied, and Computational Mathematics Pub Date : 2023-06-14 DOI:10.1117/12.2679151

Yuan Liu

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引用次数: 0

摘要

结果表明:∫∞n(𝐥𝐨𝐠𝒙)/ +𝒙𝒙=𝝅/𝟖，并选择𝐥𝐨𝐠𝒛作为对数函数的一个分支。同时,logz域的全纯:{𝒛:𝑰𝒎𝒛≥𝟎𝒂𝒏𝒅𝒛≠𝟎}。然后本研究计算如何表达残渣f𝟏+𝐳𝟐=(𝐳−𝐢)∙(𝐳+𝐢),有两个解决方案𝟏+𝐳𝟐=𝟎:𝒛𝟏=𝒊𝒂𝒏𝒅𝒛𝟐=−𝒊。函数只有一个极点出现在𝐳=𝐢。这一结果适用于更多沿曲线积分的计算。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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The integral along curves by residue theorem

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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来源期刊

Conference on Pure, Applied, and Computational Mathematics

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