英语翻译In several cases we simplify standard proof.Fox example

英语翻译
In several cases we simplify standard proof.Fox example we replace the usual tedious calculations showing that the Kelvin transform of a harmonic function is harmonic with some straightforward observations that we believe are more revealing.Another example is our proof of Bocher’s Theorem,which is more elementary than the classical proofs.
We also present material not usually covered in standard treatments of harmonic functions.The section on the Schwarz Lemma and the chapter on Bergman spaces are examples .For completeness,we include some topics in analysis that frequently slip throuth the cracks in a beginning graduate student’s curriculum ,such as real-analytic functions.
We rarely attempt to trace the history of the ideas presented in his book .Thus the absence of a reference does not imply originality on our part.
In addition to writing the text ,the authors have developed a software package to manipulate
Many of the expressions that arise in harmonic function theory.Our software package ,which uses many results from this book.,can perform symbolic calculations that would take a prohibitive amount of time if done without a computer.For example,the Poisson integral of any polynomial can be computed exactly.Appendix B explains how readers can obtain our software package free of charge.
This book has its roots in a graduate course at Michigan State University taught by one of the authors and attented by the other authors along with a number of graduate student3..The topic of harmonic functions was presented with the intention of moving on to different material.Instead ,we began to ask natural questions about harmonic functions.Lively and illuminating discussions ensued.A freewheeling approach to the course developed,answers to questions someone had raised in class or in the hallway were worked out and then presented in class( or in the hallway).Discovering mathematics in this way was a thoroughly enjoyable experience.We will consider this book a success if some of that enjoyment shines through in these pages.

在一些情况下,我们简化了标准的证明.例如我们取代通常的繁琐的计算表明,在开尔文谐波函数变换的谐波和一些直接观察,我们相信更清楚.另一个例子就是我们的定理,证明Bocher哪个更基本比经典的证明.
同时也讨论资料通常不包括在标准治疗的谐波函数.这部分许瓦兹引理及其这一章对伯格曼的空间就是例子.因为,完全性,我们在分析包括一些主题,通过这次的裂缝中频繁滑动开始研究生的课程,如real-analytic功能.
我们很少试图跟踪构想的历史在他的书中型砂的缺乏创意的参考并不意味着我们的部份上.
除了写的文字,并在此基础上开发出一个软件包来操纵
许多的表达方式,产生的谐波函数理论.我们的软件包,它使用很多成果从这卷书,能够完成象征性的计算方法,将在一禁止性的时间的数量如果做的没有电脑.例如,泊松分布的积分就可以计算任何多项式的确切位置.附录B中解释如何能得到我们的软件包读者免费的.
这本书有其根源在于研究生课程在密歇根州立大学教一名作者和其他研究者的attented随着由一定数量的毕业学生三..的主题提出了谐波函数的打算,继续向不同的材料.相反,我们就开始向自然疑问谐波函数.活泼的和富有启发性的讨论相继发生.本课程自主开发的方法,所提出的问题的答案的人在课堂上或者在走廊里都是在课堂上,然后提出了(或在走廊里).Discovering数学用这种方法是一个彻底的令人愉快的经历.我们将把这本书的成功,如果有些享受照彻的记录信息.