17Calculus Infinite Series - Fourier Series
17Calculus
Fourier Series
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What is a Fourier Series?
The main idea of Fourier Series is that we want to build an infinite series, using the basic trig functions sine and cosine, that is equivalent to a more complicated function. The series can then be manipulated more easily than the original function.
Here is a great video to get you started. He explains why we need to build these functions, goes through an example and then explains the big picture.
Dr Chris Tisdell - building functions [13min-52secs]
The solution to the practice problem at the very end of this video can be found in his free workbook found here.
video by Dr Chris Tisdell |
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How to Calculate Fourier Series
As you saw in that video, there are some basic equations required to calculate the Fourier Series. To build a Fourier Series for a function \(f(t)\) with period \(2L\), it is required that \(f(t)\) and it's derivative \(f'(t)\) be piecewise continuous on the interval \([-L,L]\).
Fourier Series Equations | |||
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Fourier Series |
\(\displaystyle{ f(t) = a_0 + \sum_{n=1}^{\infty}{ \left[ a_n \cos \frac{n \pi t}{L} + b_n \sin \frac{n \pi t}{L} \right] } }\) | ||
constants | |||
\(\displaystyle{ a_0 = \frac{1}{2L} \int_{-L}^{L}{f(t)~dt} }\) | |||
\(\displaystyle{ a_n = \frac{1}{L} \int_{-L}^{L}{f(t)\cos \frac{n\pi t}{L} ~dt} }\) | |||
\(\displaystyle{ b_n = \frac{1}{L} \int_{-L}^{L}{f(t)\sin \frac{n\pi t}{L} ~dt} }\) | |||
Knowing if the original \(f(t)\) is either even or odd can help us a lot when finding the Fourier Series. Of course, we do not require that \(f(t)\) be even or odd, but you remember from precalculus that cosine is an even function and sine is odd. So, for even functions \(b_n=0\) and for odd functions \(a_n=0\).
Deriving The Fourier Series Coefficient Equations
So where do these equations come from? In order to use those equations, you don't need to know where they come from but it is quite satisfying to understand why we divide by \(L\) or \(2L\) in the above formulas. First you need understand about orthogonality since it is an integral part of the derivation of the formulas. Here is a great video clip on orthogonality. Note: In these two video clips, he shows the formulas with \(L=\pi\). The logic is the same for any \(L>0\).
MIT OCW - Lec 15 | MIT 18.03 Differential Equations, Spring 2006 [clip 1]
video by MIT OCW |
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Okay, from that video clip, keep in mind that for two different orthogonal functions \(u(t)\) and \(v(t)\), \( \int_{-\pi}^{\pi}{ u(t)v(t)~dt } = 0 \). In this video clip, he shows the derivation of the equations to calculate the coefficients. Again, he does this specifically for \(L=\pi\) but the equations hold for any \(L\).
MIT OCW - Lec 15 | MIT 18.03 Differential Equations, Spring 2006 [clip 2]
video by MIT OCW |
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Okay, time for some practice problems.
Practice
Unless otherwise instructed, find the Fourier Series for these functions.
\(\displaystyle{ f(t) = \left\{\begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }\)
\( f(t) = f(t+2\pi)\) for all \(t\)
Problem Statement
Find the Fourier Series for \(\displaystyle{ f(t) = \left\{\begin{array}{rr} -1 & -\pi < t < 0 \\ 0 & t = 0, \pm \pi \\ 1 & 0 < t < \pi \end{array} \right. }\)
\( f(t) = f(t+2\pi)\) for all \(t\)
Solution
Dr Chris Tisdell - 1296 video solution
video by Dr Chris Tisdell |
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\(\displaystyle{f(x) = \left\{\begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }\)
with period \(2\)
Problem Statement
Find the Fourier Series for \(\displaystyle{f(x) = \left\{\begin{array}{rr} 0 & -1 \leq x \leq 0 \\ 1 & 0 < x < 1 \end{array} \right. }\)
with period \(2\)
Solution
PatrickJMT - 1297 video solution
video by PatrickJMT |
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\(\displaystyle{ f(x) = \left\{\begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }\)
\( f(x) = f(x+2) \)
Problem Statement
Find the Fourier Series for \(\displaystyle{ f(x) = \left\{\begin{array}{rr} -3 & -1 < x < 0 \\ 3 & 0 < x < 1 \end{array} \right. }\)
\( f(x) = f(x+2) \)
Solution
Dr Chris Tisdell - 1298 video solution
video by Dr Chris Tisdell |
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\(\displaystyle{ f(t) = \left\{ \begin{array}{rc} 1 & 0 < t < \pi \\ 0 & \pi < t < 2\pi \end{array} \right. }\)
Problem Statement
\(\displaystyle{ f(t) = \left\{ \begin{array}{rc} 1 & 0 < t < \pi \\ 0 & \pi < t < 2\pi \end{array} \right. }\)
Solution
Trefor Bazett - 4418 video solution
video by Trefor Bazett |
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Problem Statement
Solution
Trefor Bazett - 4419 video solution
video by Trefor Bazett |
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