I read from Fourier series a Modern Introduction Volume 1,
"D. Bernoulli, D' Alembert, Lagrange, and Euler, from about 1740 onward, were led by problems in mathematical physics to consider and discuss heatedly the possibility of representing a more or less arbitrary function $f$ with period $2\pi$ as the sum of a trigonometric series.
What works in expanding $2\pi$ functions were done by the mathematicians listed above?