傅里叶级数理论成因分析

An Exploration of the Reason Why Fourier Established the Theory of Fourier Series

在《热的解析理论》等原始文献的基础上探讨傅里叶成功创建其级数理论的原因。傅里叶对函数观念的变革性的认识为其建立级数理论扫清了障碍;受早期声乐理论中简单模式叠加观念启发,傅里叶成功求解了半无穷矩形薄片中的热传导问题,找到了建立其级数理论的思路;他通过解决圆周上离散物体的热传导问题,将三角级数展开式从无穷可微函数扩展至"任意函数";求解热传导方程的需要推动了傅里叶级数理论的建立、拓展与完善。

Based on the study of The Analytical Theory of Heat and other relevant original literature, a discussion was made of the reason why Fourier had set up the theory of Fourier series in this paper. Fourier essentially distinguished non-analytical function from the analytical function and recognized only in one interval rather than in whole domain can any function be developed by trigonometric series, which eradicated the obstacles for his setting up the theory of series; with the enlightenment of the notion of superposition of simple modes emerged from early acoustics, Fourier solved successfully the problem of communication of heat in semi-infinite strip and found the basic method of setting up the theory of Fourier series; by solving the problem of communication of heat between discrete bodies arranged around a circle, Fourier expanded trigonometric series employed in infinitely differentiable function to any function; the demand for solving equation of heat conduction stimulated Fourier to set up, expand and perfect the theory of Fourier series.