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§1 引言关于函数方程的解法和理论的研究,至今已有二百多年的历史了。这个问题可以说是在数学分析中研究得最少的问题之一。由于它的解法十分复杂,其形式千变万化,又具有极大的一般性,因此到目前为止,还没有找到它的统一理论,其一般解法知之甚少,也还没有找到函数方程的解的存在性和唯一性的条件。不仅如此,还有许多的函数方程直到现在还没有解出来。1773年法国数学蒙日为了研究“曲面论”中的问题,必须解一些函数方程,他尽量设法把这些方程化为“有限差方程”来处理;同年另一数学家拉普拉斯把蒙日的方法推广到相当广泛的一类函数方程上面去。
§ 1 Introduction The research on the solution and theory of function equations has been over two hundred years ago. This problem can be said to be one of the least studied problems in mathematical analysis. Because its solution is very complicated, its form is ever-changing, and it is extremely general. So far, it has not found its unified theory. Its general solution method knows very little and has not found the existence of the solution to the function equation. And unique conditions. Not only that, there are many functional equations that have not been solved yet. In 1773, the French mathematician Mengri had to solve some functional equations in order to study the problems in the “surface theory”. He tried to manage these equations as “limited difference equations” to deal with it; another mathematician, Laplace, put Mongolia in the same year. The method is extended to a fairly wide range of functional equations.