ABSTRACT
Throughout the 18th century, leading mathematicians became aware that power series were insufficient to represent functions, and that a different type of series was needed. Considerations of physical problems, such as the behavior of a vibrating string, suggested that their terms should be trigonometric functions. There were serious problems with trigonometric series (e.g., the convergence and the term-by-term differentiation) and very little progress was made. When Fourier showed that these series can be used to solve the problems of heat flow, there was no way back. The answers were needed and a better understanding of the fundamental concepts of calculus was necessary. Much of 19th-century mathematics has its roots in the problems associated with Fourier series.
