Step 5: Fourier sine coefficients

$$b_n = \frac{2}{L} \int_0^L f(x) \, \sin\Big(\frac{n\pi x}{L}\Big) \, dx, \quad n = 1,2,3,\dots$$ (9)

Jean-Baptiste Joseph Fourier caused a scandal with his PhD work on heat, which introduced Fourier series and challenged traditional mathematics. He later served successfully in Egypt as a scientific advisor and diplomat, under Napoleon, organizing scientific research and local improvements. Returning to France, he became a respected mathematician. Fourier’s life ended tragically in 1830 when he fell down a staircase. His work remains foundational in mathematics and physics.