I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization.
I'm happy to be part of the first Observable Ambassador's cohort.
Public
Edited
Jun 30, 2024
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A Julia set on the Riemann sphereThe Z-CurveBarnsley's fernA stochastic digraph IFS algorithmSelf-affine tilesThe TwindragonThe Eisenstein fractionsA self-affine tile with holesSelf-affine tiles via polygon mergeGolden rectangle fractalsBifurcation diagram with critical curvesThe tame twindragonIllustrations for the proof of Green's theoremNon-orientability of a Mobius stripExamples of parametric surfacesPenrose tilingThe extended unit circlePenrose three coloringNewtons's method on the Riemann sphereConic sectionsDivisor graphsThe dance of Earth and VenusIterating multiples of the sine functionBorderline fractalsSelf-similar intersectionsBox-counting dimension examplesMandelbrot by dimensionInverse iteration for quadratic Julia setsInteger Apollonian PackingsIllustrations of two-dimensonal heat flowThe logistic bifurcation locusThe eleven unfoldings of the cubeA unimodal function with fractal level curvesGreen's theorem and polygonal areaThe geometry and numerics of first order ODEsThe xxx^xxx-spindleAnimated beatsRauzy FractalsHilbert's coordinate functionsPluckNot PiDrum strikeThe Koch snowflakeFractalized squareA Taylor series about π/4\pi/4π/4PlotX3D HyperboloidA PlotX3D animationModular arithmetic in 5th grade artSimple S-I-R ModelThe Poisson KernelPoly-gasketsClassification of 2D linear systems via trace and determinantJulia sets and the Mandelbrot setWater wavesDisks for a solid of revolutionOrbit detection for the Mandelbrot setTracing a path on a spherePlot for mathematiciansFunctions of two variablesPartial derivativesDijkstra's algorithm on an RGGGradient ascentUnfolding polyhedraTangent plane to a level surfaceA strange discontinuityExamples of level surfacesMcMullen carpetsHills and valleysThe definition of ⇒Double and iterated integralsMST in an RGGTrees are bipartiteFractal typesettingd3.hierarchy and d3.treeK23 is PlanarPolar CoordinatesParametric region generatorParametric Plot 2DContour plotsGreedy graph coloringGraph6A few hundred interesting graphsThe Kings ProblemFirst order, autonomous systems of ODEsRunge-Kutta for systems of ODEs
Fourier Series
Also listed in…
Teaching PDEs
// // Symbolic solution is gone since server has been shut down.
//
// symbolic_display = {
// let display = d3.create("div");
// display.append("h3").text("Symbolic solution");
// let text = "Fetching symbolic solution...";
// let temp_display = display.append("div").attr("class", "temp").text(text);
// let interval_id = setInterval(function () {
// text = text + ".";
// temp_display.text(text);
// }, 800);
// let f = f_text;
// let L = LL;
// let fEncoded = encodeURIComponent(f);
// let LEncoded = encodeURIComponent(L);
// let base_url = "https://cgi.marksmath.org/cgi-bin/fourier_series.py?";
// let query = `f=${fEncoded}&L=${LEncoded}&type=${type}`;
// let url = base_url + query;
// fetch(url).then(async function (r) {
// let result = await r.json();
// if (result.success) {
// let type_text, lower_bound;
// if (type == "full") {
// type_text = "full Fourier";
// lower_bound = `-${result.L}`;
// } else if (type == "sine") {
// type_text = "Fourier sine";
// lower_bound = "0";
// } else if (type == "cosine") {
// type_text = "Fourier cosine";
// lower_bound = "0";
// }
// clearInterval(interval_id);
// display.select(".temp").remove();
// let container = display.append("div").attr("class", "container");
// container.append("div").text("The function,");
// container.append(() => tex.block`f(x) = ${result.f}`);
// let comment = container.append("div");
// comment
// .append("span")
// .text(`has the following ${type_text} series over `);
// comment.append(() => tex`[${lower_bound},${result.L}]`);
// comment.append("span").text(":");
// container.append(() => tex.block`f(x) \sim ${result.s}.`);
// } else {
// clearInterval(interval_id);
// display.select(".temp").remove();
// let container = display.append("div").attr("class", "container");
// if (result.error == "timeout") {
// container.html("Computation timed out.");
// } else {
// container.html(
// "There's been a problem computing the symbolic solution."
// );
// }
// }
// });
// return display.node();
// }
// Compile the function specified in the input
f = {
let f_compiled = math.compile(f_text);
return x => f_compiled.evaluate({ x: x });
}
// Maximum number of terms
N = 100
// Numeric length
mutable L = math.evaluate(LL)
// Aproximate max for plotting purposes
fmax = {
let xmin;
if (type == 'full') {
xmin = -L;
} else {
xmin = 0;
}
let fmax = d3.max(
d3
.range(xmin, L, L / 5)
.map(function(x) {
return minimize(x => -f(x), {
lowerBound: x,
upperBound: x + L / 5
});
})
.map(f)
);
return fmax;
}
// Aproximate min for plotting purposes
fmin = {
let xmin;
if (type == 'full') {
xmin = -L;
} else {
xmin = 0;
}
let fmin = d3.min(
d3
.range(xmin, L, L / 5)
.map(function(x) {
return minimize(x => f(x), {
lowerBound: x,
upperBound: x + L / 5
});
})
.map(f)
);
return fmin;
}
function fourier_series(x, n) {
let sine_portion = d3.sum(
d3
.range(n)
.map(k => fc.sine_coefficients[k] * Math.sin((k * Math.PI * x) / L))
);
let cosine_portion = d3.sum(
d3
.range(n)
.map(k => fc.cosine_coefficients[k] * Math.cos((k * Math.PI * x) / L))
);
return sine_portion + cosine_portion;
}
fc = fourier_coefficients(f)
function fourier_coefficients(f) {
let sine_coefficients;
let cosine_coefficients;
if (type == 'full') {
sine_coefficients = d3
.range(N)
.map(
n =>
int(
x => f(x) * Math.sin((n * Math.PI * x) / L),
-L,
L,
1e-10,
10,
5
) / L
);
cosine_coefficients = d3
.range(N)
.map(
n =>
int(
x => f(x) * Math.cos((n * Math.PI * x) / L),
-L,
L,
1e-10,
10,
5
) / L
);
cosine_coefficients[0] = cosine_coefficients[0] / 2;
} else if (type == 'sine') {
sine_coefficients = d3
.range(N)
.map(
n =>
int(
x => 2 * f(x) * Math.sin((n * Math.PI * x) / L),
0,
L,
1e-10,
10,
5
) / L
);
cosine_coefficients = Array.from({ length: N }, x => 0);
} else if (type == 'cosine') {
cosine_coefficients = d3
.range(N)
.map(
n =>
int(
x => 2 * f(x) * Math.cos((n * Math.PI * x) / L),
0,
L,
1e-10,
10,
5
) / L
);
cosine_coefficients[0] = cosine_coefficients[0] / 2;
sine_coefficients = Array.from({ length: N }, x => 0);
}
return {
sine_coefficients: sine_coefficients,
cosine_coefficients: cosine_coefficients
};
}
graph_style = html`<style>
g.graph > path {
stroke-width: 3px;
stroke: path
}
</style>`
d3 = require('d3-selection@2', 'd3-array@2')
import { adaptiveSimpson as int } from '@rreusser/integration'
import { text, radio, select } from "@jashkenas/inputs"
import { Range, Text, Button, Toggle } from "@observablehq/inputs"
functionPlot = require("function-plot@1/dist/function-plot")
math = require('mathjs@9')
minimize = require('https://bundle.run/minimize-golden-section-1d@3.0.0')
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