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首页> 外文学位 >A HISTORICAL DEVELOPMENT OF THE LAPLACE TRANSFORM IN MODERN OPERATIONAL CALCULUS WITH APPLICATIONS TO MATHEMATICS, PHYSICS, AND TECHNOLOGY.
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A HISTORICAL DEVELOPMENT OF THE LAPLACE TRANSFORM IN MODERN OPERATIONAL CALCULUS WITH APPLICATIONS TO MATHEMATICS, PHYSICS, AND TECHNOLOGY.

机译：现代操作计算中Laplace变换的历史发展及其在数学，物理和技术中的应用。

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摘要

This thesis is a source book for the teacher of undergraduate mathematics and engineering, and contains: (1) A short biography of Oliver Heaviside and a natural development of the Operational Calculus with examples. This can serve as supplementary material to any course at the calculus or elementary differential equation level, and to any basic engineering course concerned with transients in mechanical and electrical systems. (2) An algebraic structure of the Laplace transform in relation to Heaviside's Operational Calculus based on Arthur Erdelyi's Operational Calculus and Generalized Functions and Jan Mikusinski's Operational Calculus. Erdelyi has given us an algebraic structure of the Operational Calculus in general, including some background material for the Laplace Transform. His approach is that of Mikusinski's, and is a fine example of connections between analysis and algebra. This work of Arthur Erdelyi, presented by him as Mikusinski's theory of convolution quotients, is interpreted as applied solely to the Laplace transform. This then is expanded to an extension of the Laplace transform to a class of mathematical functions called "locally integrable" which, in general, are not Laplace transformable via complex variable theory. Included in this algebra is a treatment of the Delta function and other impulse functions.;Included in this discussion of Heaviside's contributions is an account of the mathematical justification of Heaviside's methods by Bromwich who used contour integration techniques to justify Heaviside's results, and of Carson's introduction of the Laplace transform as a mathematical basis for most of Heaviside's methods in the area of the operational calculus. Reference material is presented which contains Laplace transform examples and comparisons with other integral transforms.;Although there are a number of mathematical justifications of the operational calculus available, it is this writer's view that none are as practical and as easy to use as Mikusinski's theory of convolution quotients as presented by Erdelyi and incorporated into the thesis of the writer. Furthermore, as presented in this thesis, Laplace theory and Mikusinski's theory are shown to be completely compatible with each other, with the latter viewed as an extension of the former, resulting in an approach to the operational calculus that increases the utilitarian value of both theories.;At the end of this thesis are suggestions as to where and how various sections of it can be used with no additional mathematical preparation required on the part of the undergraduate student in typical existing undergraduate courses in mathematics and engineering.;Most undergraduate teachers of elementary differential equations are aware of the Laplace transform based operational calculus as used by scientists and engineers to solve linear differential equations. However, they may not be familiar with Heaviside's contributions in this area, nor familiar with Mikusinski's theory of convolution quotients. This thesis is written with this teacher in mind, and suggestions offered in the use of this thesis as a source book.
机译：本论文是面向大学数学和工程学教师的一本教材，其中包括：（1）奥利弗·海维赛德（Oliver Heaviside）的简短传记和《算术》的自然发展，并附有实例。它可以作为微积分或基本微分方程级的任何课程以及与机械和电气系统中的瞬变有关的任何基础工程课程的补充材料。 （2）基于亚瑟·埃尔德利（Arthur Erdelyi）的运算微积分和广义函数以及扬·米库辛斯基（Jan Mikusinski）的运算微积分，关于拉维斯变换的代数结构与Heaviside的运算微积分。 Erdelyi总体上给了我们算术演算的代数结构，包括有关拉普拉斯变换的一些背景材料。他的方法是Mikusinski的方法，并且是分析和代数之间联系的一个很好的例子。亚瑟·埃尔德利（Arthur Erdelyi）提出的这部著作作为米库辛斯基的卷积商理论，被解释为仅适用于拉普拉斯变换。然后将其扩展为拉普拉斯变换的扩展，扩展为一类称为“局部可积分”的数学函数，这些函数通常不能通过复杂变量理论进行拉普拉斯变换。在此代数中包括对Delta函数和其他脉冲函数的处理。在对Heaviside贡献的讨论中，包括了Bromwich用轮廓积分技术证明Heaviside的结果对Heaviside的方法的数学论证以及对Carson的介绍。在运算演算领域中，Laplace变换作为Heaviside大部分方法的数学基础。提供了参考资料，其中包含拉普拉斯变换示例以及与其他积分变换的比较。；尽管有许多运算学说的数学依据，但笔者认为，没有一种方法能像Mikusinski的理论一样实用且易于使用。由Erdelyi提出并纳入作者论文的卷积商。此外，如本论文所述，拉普拉斯理论和米库辛斯基理论被证明是彼此完全兼容的，后者被视为对前者的扩展，从而导致了一种可操作的演算方法，从而增加了两种理论的功利价值。在本论文的最后，提出了在不需额外的数学准备的情况下，可以在何处以及如何使用本课程的各个部分的建议，这些课程在现有的典型的数学和工程学本科课程中，对于本科生而言是不需要的。基本微分方程了解科学家和工程师用来求解线性微分方程的基于Laplace变换的运算。但是，他们可能不熟悉Heaviside在该领域的贡献，也不熟悉Mikusinski的卷积商理论。本论文是在考虑到这位老师的前提下编写的，并提供了有关使用本论文作为参考书的建议。

著录项

  • 作者

    O'BRIEN, THOMAS DANIEL.;

  • 作者单位

    Teachers College, Columbia University.;

  • 授予单位 Teachers College, Columbia University.;
  • 学科 Education Mathematics.
  • 学位 Educat.D.
  • 年度 1981
  • 页码 280 p.
  • 总页数 280
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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