# Definition:Fourier Series/Historical Note

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## Historical Note on Fourier Series

Despite the fact that the Fourier series bears the name of Joseph Fourier, they were first studied by Leonhard Paul Euler.

Fourier himself made considerable use of this series during the course of his analysis of the behaviour of heat.

The first person to feel the need for a careful study of its convergence was Augustin Louis Cauchy.

In $1829$, Johann Peter Gustav Lejeune Dirichlet gave the first satisfactory proof about the sums of Fourier series for certain types of function.

The criteria set by Dirichlet were extended and generalized by Riemann in his $1854$ paper *Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe*.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.24$: Fourier ($\text {1768}$ – $\text {1830}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($\text {1789}$ – $\text {1857}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($\text {1805}$ – $\text {1859}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)