{"title":"欧拉三叉方程级数解的证明","authors":"Fei Wang","doi":"10.1145/3055282.3055284","DOIUrl":null,"url":null,"abstract":"In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form <i>x</i><sup><i>α</i></sup> − <i>x</i><sup><i>β</i></sup> = (<i>α</i> − <i>β</i>)<i>vx</i><sup><i>α</i>+<i>β</i></sup>. In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved in the paper. Euler's paper gave the first series expanion for the so-called Lambert W function. In this work, we briefly review Euler's results and give a proof to modern standards of rigor of the series solution of Lambert's transcendental equation.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"136-144"},"PeriodicalIF":0.0000,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Proof of a series solution for euler's trinomial equation\",\"authors\":\"Fei Wang\",\"doi\":\"10.1145/3055282.3055284\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form <i>x</i><sup><i>α</i></sup> − <i>x</i><sup><i>β</i></sup> = (<i>α</i> − <i>β</i>)<i>vx</i><sup><i>α</i>+<i>β</i></sup>. In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved in the paper. Euler's paper gave the first series expanion for the so-called Lambert W function. In this work, we briefly review Euler's results and give a proof to modern standards of rigor of the series solution of Lambert's transcendental equation.\",\"PeriodicalId\":7093,\"journal\":{\"name\":\"ACM Commun. Comput. Algebra\",\"volume\":\"1 1\",\"pages\":\"136-144\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2017-02-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Commun. Comput. Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3055282.3055284\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Commun. Comput. Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055282.3055284","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Proof of a series solution for euler's trinomial equation
In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form xα − xβ = (α − β)vxα+β. In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved in the paper. Euler's paper gave the first series expanion for the so-called Lambert W function. In this work, we briefly review Euler's results and give a proof to modern standards of rigor of the series solution of Lambert's transcendental equation.
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