Complex Variables and Analytic Functions: An Illustrated Introduction最新文献

Chapter 7: Elliptic Functions 第七章:椭圆函数

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch7

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Chapter 4: Introduction to Complex Integration 第4章:复杂集成简介

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch4

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Chapter 12: Steepest Descent for Approximating Integrals 第十二章:逼近积分的最陡下降法

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch12

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Chapter 11: Special Functions Defined by ODEs 第11章:ode定义的特殊函数

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch11

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Chapter 6: Gamma, Zeta, and Related Functions 第六章:Gamma, Zeta和相关函数

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch6

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Chapter 2: Functions of a Complex Variable 第二章:复变量的函数

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch2

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Chapter 3: Analytic Continuation 第三章:解析延拓

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch3

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Chapter 10: Wiener–Hopf and Riemann–Hilbert Methods 第十章:Wiener-Hopf和Riemann-Hilbert方法

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch10

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Chapter 1: Complex Numbers 第1章:复数

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch1

{"title":"Chapter 1: Complex Numbers","authors":"","doi":"10.1137/1.9781611975987.ch1","DOIUrl":"https://doi.org/10.1137/1.9781611975987.ch1","url":null,"abstract":"Preliminary considerations Historically, complex numbers were introduced to solve algebraic equations. It was observed that the algebraic equation x 2 + 1 = 0 possesses no real solutions and the notation ± √ −1 was used to represent the roots. It was the year 1545 when the mathematicians Giro-lamo Cardano 1 , Ferro Tartaglia 2 and Rafel Bombeli 3 first used the square root of negative numbers to represent roots of equations. In 1797 a Norwegian Surveyor Wessel 4 reported to the Danish Academy of Science on the geometric interpretation of complex numbers of the form z = a + ib. Later in 1806 Robert Argand 5 also produced an essay on the geometric interpretation of complex numbers. The Swiss mathematician Leonard Euler 6 was the first to introduce the symbol i to represent the imaginary component of a number. In 1831 Carl Fredrich Gauss 7 formalized the work of Wessel and Argand and introduced numbers z = x + iy where i is a symbol used to represent a pure imaginary number having the property that i 2 = −1. Numbers of the form z = x + iy, where x and y are real numbers, were called complex numbers and many mathematicians have contributed to the development of the theory of complex numbers and functions associated with these numbers. In particular, much of the early work in the complex domain was done by the mathematicians Cauchy 8 , Weierstrass 9 and Riemann 10. For the last two hundred years many applications of complex numbers and complex functions have been developed in the areas of science and engineering. For example, the study areas of mappings, integration, differentiation, solutions of algebraic equations, solutions of ordinary differential equations, solutions of partial differential equations , summation of series, sequences, fractals and potential theory are just a few of the many application areas where one can find complex variable theory employed. You will find examples from many of these application areas presented throughout this textbook.","PeriodicalId":121211,"journal":{"name":"Complex Variables and Analytic Functions: An Illustrated Introduction","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114298153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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Chapter 9: Transforms 第九章:变换

Complex Variables and Analytic Functions: An Illustrated Introduction Pub Date : 2019-01-01 DOI: 10.1137/1.9781611975987.ch9

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