Examples of Fourier Series
Leif Mejlbro
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Examples of Fourier series Calculus 4c-1
Download free ebooks at bookboon.com Examples of Fourier series – Calculus 4c-1 © 2008 Leif Mejlbro & Ventus Publishing ApS ISBN 978-87-7681-380-2
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Contents
Introduction 5
1. Sum function of Fourier series 6
2. Fourier series and uniform convergence 62
3. Parseval’s equation 101
4. Fourier series in the theory of beams 115
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4 Examples of Fourier series Introduction
Introduction
Here we present a collection of examples of applications of the theory of Fourier series. The reader is also referred to Calculus 4b as well as to Calculus 3c-2.
It should no longer be necessary rigourously to use the ADIC-model, described in Calculus 1c and Calculus 2c, because we now assume that the reader can do this himself.
Even if I have tried to be careful about this text, it is impossible to avoid errors, in particular in the first edition. It is my hope that the reader will show some understanding of my situation.
Leif Mejlbro 20th May 2008
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5 Examples of Fourier series Sum function of Fourier series
1 Sum function of Fourier series
A general remark. In some textbooks the formulation of the main theorem also includes the unnecessary assumption that the graph of the function does not have vertical half tangents. It should be replaced by the claim that f ∈ L2 over the given interval of period. However, since most people only know the old version, I have checked in all examples that the graph of the function does not have half tangents. Just in case ... ♦
n Example 1.1 Prove that cos nπ =(−1) , n ∈ N0. Find and prove an analogous expression for π π cos n and for sin n . 2 2 (Hint: check the expressions for n =2p, p ∈ N0,andforn =2p − 1, p ∈ N).
Pi/2 1 (cos(t),sin(t))
0.5
-Pi 0
–1 –0.5 0.5 1
–0.5
–1 –3/2*Pi
One may interpret (cos t, sin t) as a point on the unit circle. The unit circle has the length 2π, so by winding an axis round the unit circle we see that nπ always lies in (−1, 0) [rectangular coordinates] for n odd, and in (1, 0) for n even. It follows immediately from the geometric interpretation that
cos nπ =(−1)n.
We get in the same way that at π 0forn ulige, cos n = 2 (−1)n/2 for n lige, and π (−1)(n−1)/2 for n ulige, sin n = 2 0forn lige.
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6 Examples of Fourier series Sum function of Fourier series
Example 1.2 Find the Fourier series for the function f ∈ K2π, which is given in the interval ]−π,π] by 0 for − π and find the sum of the series for t =0. 1 –4 –2 2 4 x 1 ∗ Obviously, f(t) is piecewise C without vertical half tangents, so f ∈ K2π. Then the adjusted function f ∗(t) is defined by f(t)fort = pπ, p ∈ Z, f ∗(t)= 1/2fort = pπ, p ∈ Z. The Fourier series is pointwise convergent everywhere with the sum function f ∗(t). In particular, the sum of the Fourier series at t =0is 1 f ∗(0) = , (the last question). 2 You’re full of energy and ideas. And that’s © UBS 2010. All rights reserved. just what we are looking for. Looking for a career where your ideas could really make a difference? UBS’s Graduate Programme and internships are a chance for you to experience for yourself what it’s like to be part of a global team that rewards your input and believes in succeeding together. Please click the advert Wherever you are in your academic career, make your future a part of ours by visiting www.ubs.com/graduates. www.ubs.com/graduates Download free ebooks at bookboon.com 7 Examples of Fourier series Sum function of Fourier series The Fourier coefficients are then 1 π 1 π a0 = f(t) dt = dt =1, π −π π 0