函数方程及其如何求解外文翻译资料

函数方程及其如何求解

Christopher G. Small University of Waterloo

摘要：函数方程类似代数方程,它是关于未知函数而不是未知数的方程。G.Small在本书详细介绍了初等函数方程的各种常见解题技巧与方法。本书之目的是为大学以及中学水平的学生学习函数方程提供一本初等函数方程求解的入门读物。

关键词：函数方程； 解决方法

背景介绍

1.1初步评价

对于高中时期的代数，我们学习代数方程涉及一个或多个未知的实数。函数方程就像代数方程,是数量未知的函数而不是实数。这本书是关于初等函数方程的各种常见解题技巧与方法。函数方程在数学竞赛中经常出现。本书之目的是为大学以及中学水平的学生学习函数方程提供一本初等函数方程求解的入门读物。

在这一章,我们研究的是一种相当广泛的函数方程，而不是集中在这类方程的解决方案 ——一个为以后章节的主题——我们知晓函数方程产生于数学研究中。我们进入主题是基本的，但也不完全是有根据的。

1.2尼克尔·奥雷斯姆

数学家处理函数方程的时间比正式的数学法则出现的时间要早很多。早期的函数方程的例子可以追溯到至14世纪数学家尼克尔·奥雷斯姆，他们的工作提供了一个间接的定义线性函数通过函数方程。尼克尔·奥雷斯姆生于1323年,死于1382年。从这些日期的角度来看,我们应该注意到,可怕的黑死病,横扫欧洲杀死可能多达三分之一的人口,发生在14世纪中期。虽然黑死病的起源尚不清楚,我们知道到1347年12月,它到达了西方地中海港口的西西里,然后到达撒丁岛,然后到了港口城市马赛。到达巴黎是在1348年的春天,在该国大部分地区进行了传播。

1348年也是具有数学意义的一年,因为正是今年尼克尔·奥雷斯姆记录了在巴黎大学的奖学金持有者的列表。因此看来尼克尔·奥雷斯姆在巴黎大学学习的时间是在1340年代早期到黑死病的时间抵达这个城市本身。今天,也许我们可能会想知道,在面对这种灾难性的疾病,学者如奥雷斯姆的工作成果如何得以蓬勃发展。然而,黑死病在奥雷斯姆一代是更具破坏性的。他和他的同事们已经完成了大部分的教育的时候,明显感受到了黑死病的破坏性影响。到1355年,奥雷斯姆已经获得了神学硕士学位。之后,他很快在纳瓦拉大学就被任命为大师,巴黎大学的学院之一,成立于1304年。尼克尔·奥雷斯姆可以说是最伟大的欧洲14世纪的数学家。他死于1382年的卡昂。

虽然在他所处的中世纪的世界里,亚里士多德的著作是占主导地位，我们现在称之为自然，奥雷斯姆的学术工作预示着后来作家的文艺复兴和启蒙运动时期,谁脱离亚里士多德将力学定律,通过这样做,创造经典物理学领域。1352年,奥雷斯姆主要论述统一和写道不同形状的强度,名为《Tractatus de configurationibus motuum qualitatum》。在这一重要工作,奥雷斯姆建立功能的定义两个变量之间的关系,以及(远远超过笛卡尔),一个可以表达这种关系由我们现在称之为几何图。第1部分中他写道。

因此,每一个强度可获得先后应该想象由一条直线垂直地竖立在一些点的空间或主题没有实体的事情,例如质量。

发现任何强度与强度之间的比例,同样和强度有关,找到类似的线与线之间的存在的比率,反之亦然。

中央对他的论文的想法的关于统一的运动和“匀不规则形状的运动”,后者表示粒子的运动经历等加速度。也被认为是“不同形状的不规则形状的运动”,加速本身变化。在四边形的部分质量，奥雷斯姆接替照顾定义他的制服不相似的概念(即线性)。

统一的质量是同样强烈的主题,而质量均匀不规则形状的就是如果有三分(主题)之间的距离比第一和第二之间的距离，第二和第三是比强度过剩的第一点,第二点的过剩的第二点,第三点,调用第一个3分之一的人最大的强度。

作为Aczel [1984]和Aczel与Dhombres [1985] 指出,通过定义了一个线性函数(即质量均匀不规则形状的)通过一个函数方程。在现代的术语中,我们会有三个不同的实数x，y和z,上面的段落中描述说,作为主题的三分。与x，y和z,我们有一个变量(即“强度”的质量在每个点的主题),我们可以写为f(x),f(y)和f(z)。定义的函数f是线性的,或“不规则形状的统一”

如果对于所有不同的 （1.1）

奥雷斯姆对于函数方程的定义,f是抽象的:你可以把这个代入任何函数方程方程是否满足所有可能的x,y,z。我们可以比较这个标准定义被发现在大多数介绍现代教科书说一个线性函数的形式

，对于一些a,b （1.2）

奥雷斯姆的关于方程(1.1)是一个函数方程。定义在(1.2)的是其解决方案。注意,奥雷斯姆定义的年代不允许常数线性函数关于。这是忠实于他的意图,统一不同形状的功能区别于统一的函数根据选择和 。

1.3格里高利圣文森特

在接下来的几百年,函数方程被使用着但没有再产生一般理论。圣文森特的引人注目的在这样的数学家是格雷戈里(1584 - 1667),他的工作在双曲线函数方程的隐式使用，开创了对数理论。

圣文森特的结果出现在他伟大的写于1647的一篇论文题为《Geometricum quadraturae circuli et sectionum coni》中。如果这项工作的标题看起来长,论文本身,在约1250页,是更长的时间!它处理方法计算区域和圆锥部分的属性。特别是,圣文森特显示了它是如何可能的计算面积等双曲线如图1.3所示。在现代,曲线下的面积,如一个双曲线主题通常留给集成理论。然而,圣文森特岛问题上取得了很大的进步使用纯粹的几何参数。

特别是,圣文森特的论点是基于下面的几何原理。

如果水平拉伸由给定平面区域因素,同时缩小垂直同样的因素,然后产生的地区将有一个区域,等于原来的地区。

例如,在图1.2中,我们看到,垂直和水平扩展了平面区域伸展因子2水平,垂直和缩水的2倍。第二个地区同一地区第一。

现在让我们来看看这个几何原理适用于面积双曲线。让上的阴影区域表示间隔从1到x如图1.3所示,并考虑相应的阴影区域在相同双曲线竖立在间隔从到对于任意来说。比较两个阴影区域,圣文森特指出他们不同y沿着x轴的比例因子,并通过沿y轴的比例因子。因此,这两个地区的区域必须是相同的。

阴影区域的面积与基地从到是。在此之前立即从事实下的地区从双曲线到正是通过消除该地区的从1到从该地区从1到。因此,使用圣文森特缩放参数,我们有

或者说

我们现在认识到这个方程作为独特的种类的对数函数方程。然而,理论工作链接这个函数方程对数的种类不得不等待柯西的工作。

工作曲线和计算区域,圣文森特还记得从他的论文的第二部分贡献作品几何,在那里他学习了无穷级数。与他的工作领域,彻底研究的方法,和系列,格里高利圣文森特是早期的现代方法计算和分析开拓者之一。

1.4奥古斯丁·路易·柯西

虽然尼克尔·奥雷斯姆关于的线性的定义可以被解读为一个函数方程的早期例子,它不代表一个函数方程的理论起点。函数方程的定义的日期是从柯西的工作开始的。柯西1789年生于法国巴黎,柯西的早期恰逢法国大革命。在上下文联系他的生日,我们应该回想一下,法国大革命是一般追溯到1789 - 1799年十年间,开始大约1789年攻占巴士底狱。1799年,年轻的柯西十岁的时候,拿破仑·波拿巴将军领导的政变的督政府开始一段法国的直接军事统治。柯西家人出于对保皇党人的同情,他们离开巴黎,直到1800年才返回。强大的君主者，奥古斯丁·路易·柯西后来背道而驰共和党和他早年在法国拿破仑趋势。一个杰出的数学家,柯西在数学的许多领域工作。然而,他主要是以微积分而闻名,是现代数学分析的理论公认的创始人之一。

与柯西有关的函数方程是

（1.3）

所有真正的x和y,现在被称为柯西方程。需要找到所有实值函数f满足方程(1.3)。现在读者可以立即注意到柯西方程满足任何形式的函数

，

常数是一个任意的实数。然而,我们能够找到一个简单的解决方案,这个方程只是故事的一小部分。我们还必须问同一种类型形式的全套解决方案是诸如方程(1.3)。似乎合理,这种线性函数是唯一的解决方案(1.3)。然而,这原来是真的只有一些轻微的限制强加给函数f。例如,函数的形式是唯一的解决方案(1.3)在类的函数有界区间的形式,其中。或者,它可以表明,形成的唯一类的解决方案在连续的实值函数在实线。我们研究这个方程及其在第二章中详细的解决方案。

外文文献出处：[B].Problem Books in Mathematics2007Springer

附外文文献原文

An historical introduction

1. Preliminary remarks

In high school algebra, we learn about algebraic equations involving one or more unknown real numbers. Functional equations are much like algebraic equations, except that the unknown quantities are functions rather than real numbers. This book is about functional equations: their role in contemporary mathematics as well as the body of techniques that is available for their solution. Functional equations appear quite regularly on mathematics competitions. So this book is intended as a toolkit of methods for students who wish to tackle competition problems involving functional equations at the high school or university level.

In this chapter, we take a rather broad look at functional equations. Rather than focusing on the solutions to such equations—a topic for later chapters— we show how functional equations arise in mathematical investigations. Our entry into the subject is primarily, but not solely, historical.

1. Nicole Oresme

Mathematicians have been working with functional equations for a much longer period of time than the formal discipline has existed. Examples of early functional equations can be traced back as far as the work of the fourteenth century mathematician Nicole Oresme who provided an indirect definition of linear functions by means of a functional equation. Of Norman heritage, Oresme was born in 1323 and di

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An historical introduction

1. Preliminary remarks

In high school algebra, we learn about algebraic equations involving one or more unknown real numbers. Functional equations are much like algebraic equations, except that the unknown quantities are functions rather than real numbers. This book is about functional equations: their role in contemporary mathematics as well as the body of techniques that is available for their solution. Functional equations appear quite regularly on mathematics competitions. So this book is intended as a toolkit of methods for students who wish to tackle competition problems involving functional equations at the high school or university level.

In this chapter, we take a rather broad look at functional equations. Rather than focusing on the solutions to such equations—a topic for later chapters— we show how functional equations arise in mathematical investigations. Our entry into the subject is primarily, but not solely, historical.

1. Nicole Oresme

Mathematicians have been working with functional equations for a much longer period of time than the formal discipline has existed. Examples of early functional equations can be traced back as far as the work of the fourteenth century mathematician Nicole Oresme who provided an indirect definition of linear functions by means of a functional equation. Of Norman heritage, Oresme was born in 1323 and died in 1382. To put these dates in perspective, we should note that the dreaded Black Death, which swept through Europe killing possibly as much as a third of the population, occurred around the middle of the fourteenth century. Although the origins of the Black Death are unclear, we know that by December of 1347, it had reached the western Mediterranean through the ports of Sicily, then Sardinia, then the port city of Marseilles. It reached Paris in the spring of 1348, having spread throughout much of the country.

The year 1348 is also of some significance in mathematics, because that is the year that Nicole Oresme is recorded in a list of scholarship holders at the University of Paris. Thus it appears that Oresme was studying at the University of Paris from some time in the early 1340s up to the time the Black Death arrived in the city itself. Today, perhaps, we might well wonder how, in the face of this calamitous disease, scholars such as Oresme were able to flourish. However, the Black Death was more disruptive for the generation that followed Oresme. He and his colleagues had completed much of their education by the time that the Black Deathrsquo;s devastating effects were felt. By 1355, Oresme had obtained the Master of Theology degree, and was soon thereafter appointed Grand Master at the College of Navarre, one of the colleges of the University of Paris, founded in 1304. Nicole Oresme was arguably the greatest European mathematician of the fourteenth century. He died in 1382 in Lisieux.

Although he lived in a medieval world in which the writings of Aristotle were the dominant influence on natural philosophy—what we would now call the natural sciences—Oresmersquo;s scholarly work foreshadowed the work of later writers in the Renaissance and Enlightenment periods, who broke away from Aristotle to reformulate the laws of mechanics and, by so doing, create the field of classical physics. In 1352, Oresme wrote a major treatise on uniformity and difformity of intensities, entitled Tractatus de configurationibus qualitatum et m^otuv^m. In this important work, Oresme established the definition of a functional relationship between two variables, and the idea (well ahead of Rene Descartes) that one can express this relationship geometrically by what we would now call a graph.¹ In Part 1 he wrote

Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality.

For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa.^{[1] [2]}

Of central interest in his treatise is the idea of uniform motion and “uniformly difform motion,” the latter denoting the motion of a particle undergoing uniform acceleration.^[3] Also considered was “difformly difform motion,” where the acceleration itself varied. In the section on quadrangular quality, Oresme took care to define his notion of uniform difformity (i.e., linearity) as follows.

A uniform quality is one which is equally intense in all parts of the subject, while a quality uniformly difform is one in which if any three points [of the subject line] are taken, the ratio of the distance between the first and the second to the distance between the second and the third is as the ratio of the excess in intensity of the first point over that of the second point to the excess of that of the second point over that of the third point, calling the first of those three points the one of greatest intensity.

As Aczel [1984] and Aczel and Dhombres [1985] have noted, the passage defines a linear function (i.e., a quality which is uniformly difform) through a functional equation. In modern terminology, we would have three distinct^[4] real numbers x, y, and ^, say, which are described in the passage above as three points of the subject line. Associated with x, y and ^, we have a variable (i.e., the “intensity” of the quality at each point of the subject line) which we can write as f (x), f (y), and f (z), respectively. The function f is d

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