Mary's Notes

A series of Bessel Functions of the First Kind: https://en.wikipedia.org/wiki/Bessel_function

MATLAB SCRIPT:

y = linspace (-10,100, 1000);

[x0] = besselj(0, y); [x1] = besselj(1, y); [x2] = besselj(2, y); [x3] = besselj(3, y); [x4] = besselj(4, y); [x5] = besselj(5, y); [x6] = besselj(6, y); [x7] = besselj(7, y); [x8] = besselj(8, y); [x9] = besselj(9, y); [x10] = besselj(10, y); [x11] = besselj(11, y);

plot (y, x0, 'LineWidth', 1.5, 'Color',[0.4,0,0]) %red hold on plot (y, x1, 'LineWidth', 1.5, 'Color',[0.4,0.4,0])%yellow plot (y, x2, 'LineWidth', 1.5, 'Color',[0,0.4,0])%green plot (y, x3, 'LineWidth', 1.5, 'Color',[0,0,0.4])%blue plot (y, x4, 'LineWidth', 1.5, 'Color',[0.8,0,0])%red plot (y, x5, 'LineWidth', 1.5, 'Color',[0.8,0.8,0])%yellow plot (y, x6, 'LineWidth', 1.5, 'Color',[0,0.8,0])%green plot (y, x7, 'LineWidth', 1.5, 'Color',[0,0,0.8])%blue plot (y, x8, 'LineWidth', 1.5, 'Color',[1,0,0])%red plot (y, x9, 'LineWidth', 1.5, 'Color',[1,1,0])%yellow plot (y, x10, 'LineWidth', 1.5, 'Color',[0,1,0])%green plot (y, x11, 'LineWidth', 1.5, 'Color',[0,0,1])%blue

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 + ...    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])