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Cell["3.029 Spring 2022\[LineSeparator]Lecture 23 - 04/27/2022", "Subtitle",
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Cell["Fun with Fourier Series", "Chapter",
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 "Fourier Series and Fourier Transforms are mathematical ",
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This is often a more compact notation, which allows us to take advantage of \
the dominant processes in a particular phenomenon or dataset\
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The importance of the Fourier transform in science and engineering cannot be \
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Here are some examples of its use In Materials Science and Engineering:\
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Cell["Reciprocal space and diffraction (3.010)", "Subsubitem",
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Cell["\<\
You\[CloseCurlyQuote]ve already seen the formal definitions of Fourier Series \
and Fourier Transforms last semester in 3.019\
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We will quickly review the definitions, but will try and offer an alternative \
interactive exposition\
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Jean-Baptiste Joseph Fourier introduced the concept of a Fourier Series in \
1807, while working on solving the heat equation\
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The solution is in-fact identical to had we superimposed our two solutions \
independently\
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it means we can write the down the time-evolution of all* possible initial \
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Mathematica\[CloseCurlyQuote]s default convention for \
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Cell[CellGroupData[{

Cell["\[OpenCurlyQuote]Fourier\[CloseCurlyQuote] Epicycles", "Section",
 CellChangeTimes->{{3.860049865113401*^9, 3.8600498680489197`*^9}, {
  3.860055129234106*^9, 3.860055130738552*^9}, {3.860055165513565*^9, 
  3.860055172257434*^9}},ExpressionUUID->"30dde6b2-348a-4e60-ace3-\
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Cell["\<\
Let\[CloseCurlyQuote]s see how far we can push this idea of \
\[OpenCurlyDoubleQuote]approximating periodic curves\[CloseCurlyDoubleQuote] \
using sine and cosine terms\
\>", "Item",
 CellChangeTimes->{{3.860055196473669*^9, 3.8600552010847263`*^9}, {
  3.860055350932167*^9, 
  3.8600553896700287`*^9}},ExpressionUUID->"e26ad6cd-b98f-49b0-ace5-\
04110635fb63"],

Cell[CellGroupData[{

Cell[TextData[{
 "First, note that it\[CloseCurlyQuote]s more convenient to work in complex \
notation, using Euler\[CloseCurlyQuote]s formula\[LineSeparator]\
\[LineSeparator]",
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In addition to the sine-cosine relation to exponentials, the above figure \
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