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I wanted to post a little bit of what I’ve learned in physics so far this year, partly because I’m proud of it and partly because it looks neat. This was all programmed in MATLAB, except for the cat. Pictured: Fourier Approximation of x³, hold off and hold on because hold on is rainbow, and a simple pendulum. (...and Willow standing on some homework.)Â
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Fourier Approximations for simple x^n functions on [-1, 1]
https://en.wikipedia.org/wiki/Fourier_series
MATLAB SCRIPT:
%Lots of thanks too: %https://www3.nd.edu/~nancy/Math30650/Matlab/Demos/fourier_series/fourier_series.html
%******Magic Fourier Solver****** syms k L n x evalin(symengine,'assume(k,Type::Integer)'); %A_n a = @(f,x,k,L) int(f*cos(k*pi*x/L)/L,x,-L,L); %B_n b = @(f,x,k,L) int(f*sin(k*pi*x/L)/L,x,-L,L); %nth partial sum fs = @(f,x,n,L) a(f,x,0,L)/2 + ... Â Â symsum(a(f,x,k,L)*cos(k*pi*x/L) + b(f,x,k,L)*sin(k*pi*x/L),k,1,n);
%******FUNCTIONS(s) GO HERE******* f1 = x;
%******Plotter****** figure
p1 = ezplot(fs(f1,x,1,1)); hold on p2 = ezplot(fs(f1,x,2,1)); p3 = ezplot(fs(f1,x,3,1)); p4 = ezplot(fs(f1,x,4,1)); p5 = ezplot(fs(f1,x,5,1)); p6 = ezplot(fs(f1,x,6,1)); p7 = ezplot(fs(f1,x,7,1)); p8 = ezplot(fs(f1,x,8,1)); p9 = ezplot(fs(f1,x,9,1)); p10 = ezplot(fs(f1,x,100,1)); p11 = ezplot(fs(f1,x,1000,1));
%Graph Customization set(p1, 'LineWidth', 1.2, 'Color',[0.8,0,0.2]) set(p2, 'LineWidth', 1.2, 'Color',[1,0,0]) set(p3, 'LineWidth', 1.2, 'Color',[1,0.5,0]) set(p4, 'LineWidth', 1.2, 'Color',[1,1,0]) set(p5, 'LineWidth', 1.2, 'Color',[0,1,0]) set(p6, 'LineWidth', 1.2, 'Color',[0,0.6,0.2]) set(p7, 'LineWidth', 1.2, 'Color',[0,0.8,0.8]) set(p8, 'LineWidth', 1.2, 'Color',[0,0,1]) set(p9, 'LineWidth', 1.2, 'Color',[0.2,0,0.8]) set(p10, 'LineWidth', 1.2, 'Color',[0.5,0,0.5]) set(p11, 'LineWidth', 1.2, 'Color',[0,0,0])
axis([-1 1 -1.5 1.5]) grid on legend('n=1', 'n=2', 'n=3', 'n=4', 'n=5', 'n=6', 'n=7', 'n=8', 'n=9', 'n=100', 'n=1000', 'Location', 'best') xlabel('x'); ylabel('y'); title({'Fourier Approximations for f(x) = (Function goes here)'});
Be honest; you've always wanted to do this.
On Fourier Approximation
A Quantized Confusion Original -- Fourier Analysis Compared to Baking!
Different frequencies of waves are like ingredients in a recipe, where "how much" of each ingredient determines what sort of final product you will create.
Whether these ingredients are the stationary states of the quantum harmonic oscillator, or harmonics of an instrument--Fourier Analysis is an infinite concoction of known functions.
