Joseph Fourier

1. Fourier Series

Introduced in 1807, this idea was initially controversial for claiming arbitrary functions could be represented by trigonometric series.

$$ f(x) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos nx + b_n \sin nx\right) $$

2. Fourier Transform

Developed from Fourier’s heat theory, this transform later became essential to physics, engineering, and signal processing.

$$ \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i \xi x}dx $$

3. Heat Equation

Fourier formulated this equation in 1807 while studying heat diffusion, laying groundwork for modern partial differential equations.

$$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u $$