Fourier, the Animated Series - We’ve seen many graphical and animated explainers for the Fourier series. We suppo... - https://hackaday.com/2024/06/05/fourier-the-animated-series/ #fourierseries #visualization #mathematics #waveforms #science

Fourier, the Animated Series https://hackaday.com/2024/06/05/fourier-the-animated-series/ #fourierseries #visualization #mathematics #waveforms #Science

The Fourier transform (FT), explained in one sentence: https://blog.revolutionanalytics.com/2014/01/the-fourier-transform-explained-in-one-sentence.html
$$\boxed{{\displaystyle \hat{f}(\xi )={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi \xi t}f(t)\mathrm{d}t}}}$$

Discrete Fourier transform (DFT):
$${\displaystyle x_n=\sum _{k=0}^{N-1}{X}_k\cdot e^{-i2\pi \frac{n}{N}k}}$$

Inverse transform:
$${\displaystyle X_k=\frac{1}{N}\sum _{n=0}^{N-1}{x}_n\cdot e^{i2\pi \frac{n}{N}k}}$$

John von Neumann once claimed, "with 4 parameters, I can fit an elephant, and with 5, I can make him wiggle his trunk."
$$x(t)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}(A_k^x\cos (kt)+B_k^x\sin (kt))}$$
$$y(t)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}(A_k^y\cos (kt)+B_k^y\sin (kt))}$$
Here's a paper proving that von Neumann's claim is valid! https://aapt.scitation.org/doi/10.1119/1.3254017
#Neumann #JohnVonNeumann #VonNeumann #FourierSeries #parameters #complexparameters #parametrization #mathematics #maths

Interactive Demo Shows the Power of Fourier Transforms - When it comes to mathematics, the average person can probably get through most of life well enough... more: https://hackaday.com/2019/04/10/interactive-demo-shows-the-power-of-fourier-transforms/ #fouriertransform #internethacks #softwarehacks #decomposition #fourierseries #epicycles #encoding #fourier #news #jpeg #mpeg #sine #svg