Fourier Analysis with Donny Lee

Video Lectures

Displaying all 17 video lectures.
Lecture 1
Definition of the Fourier Series
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Definition of the Fourier Series
In this video lesson, Math Instructor Donny Lee begins a study on the work of Joseph Fourier (1768-1830) with the definition of the Fourier Series: A way of expressing functions as infinite sums or integrals or trigonometry functions.
Lecture 2
Fourier Series of a 'Broken' Function
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Fourier Series of a 'Broken' Function
In this video lesson, Math Instructor Donny Lee uses the definitions from the previous lesson to give an example of how you write a Fourier Series from a graph of a function, a 'broken' function in this case.
Lecture 3
Fourier Series of Function on [-L, L]
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Fourier Series of Function on [-L, L]
In this video lesson, Math Instructor Donny Lee uses a simple substitution, to extend the definition of a Fourier Series of a function, this time integrable from -L to L.
Lecture 4
Example of Fourier Series on [-L, L]
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Example of Fourier Series on [-L, L]
In this video lesson, Math Instructor Donny Lee looks at a simple example, this time a function integrable on [-3, 3].
Lecture 5
Fourier Series On Odd and Even Functions
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Fourier Series On Odd and Even Functions
In this video lesson, Math Instructor Donny Lee takes advantage of certain properties of odd and even functions to simplify the work in finding the Fourier Series.

Note: Error in the video, for odd functions, -f(x)=f(-x).
Lecture 6
Fourier Series of x^2 [may not = f(x)]
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Fourier Series of x^2 [may not = f(x)]
In this video lesson, Math Instructor Donny Lee uses the even property of a function, to find the Fourier Series of x^2 on [-3, 3] which serves as an example of why certain functions do not need to agree with its Fourier Series. A Convergence Theorem is needed.
Lecture 7
Recap On Piecewise Continuous Functions
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Recap On Piecewise Continuous Functions
In this video lesson, Math Instructor Donny Lee will revise some basic definitions of limits and derivatives of piecewise continuous functions, which are needed in learning the convergence theorem of Fourier Series.
Lecture 8
Convergence Theorem of a Fourier Series
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Convergence Theorem of a Fourier Series
In this video lesson, Math Instructor Donny Lee discusses the Convergence Theorem of a Fourier Series, a very important theorem in Fourier Analysis.
Lecture 9
Convergence of Fourier Series of f(x)=2x
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Convergence of Fourier Series of f(x)=2x
In this video lesson, Math Instructor Donny Lee uses the Convergence Theorem, and looks at a simple function: f(x) = 2x on [-pi, pi].
Lecture 10
Convergence of a 'broken' Function
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Convergence of a 'broken' Function
In this video lesson, Math Instructor Donny Lee looks at a slightly more complicated example, and discusses "Where does the Fourier Series of a 'broken' function converge to?".
Lecture 11
Graphs of Various Fourier Series
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Graphs of Various Fourier Series
In this video lesson, Math Instructor Donny Lee looks at the graphs of various Fourier Series. To illustrate the series, Donny Lee takes the Nth partial sum. It is also here where we notice some interesting behavior of some Fourier Series.
Lecture 12
Contrasting Power and Fourier Series
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Contrasting Power and Fourier Series
In this video lesson, Math Instructor Donny Lee studies the convergence theorem, and compares that with another convergence theorem, that being the one for the Power Series.
Lecture 13
Fourier Cosine Series
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Fourier Cosine Series
In this video lesson, Math Instructor Donny Lee talks about the Fourier Series of functions which are integrable on [0, L].
Lecture 14
Fourier Sine Series
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Fourier Sine Series
In this video lesson, Math Instructor Donny Lee teaches another way of writing the Fourier Series of functions which are integrable on [0, L], which is to do a half range expansion, this time with sine terms.
Lecture 15
Half Range Expansions of f(x)=e^2x
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Half Range Expansions of f(x)=e^2x
In this video lesson, Math Instructor Donny Lee uses the techniques taught for getting the half range expansion of a function with either cosine or sine terms, and looks at f(x)=e^2x on [0, 1].
Lecture 16
Periodic Functions
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Periodic Functions
In this video lesson, Math Instructor Donny Lee extends the basic ideas of Fourier Series and looks at periodic functions, which repeat themselves after a period T.
Lecture 17
Phase Angle Form of a Function
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Phase Angle Form of a Function
In this video lesson, Math Instructor Donny Lee will find the Fourier Series of a periodic function g(x), write it in its phase angle form and glimpse at its amplitude spectrum.