Suppose now that we have a periodic function f(x) with period 2 such that f(x) = { x + 1 if − π ≤ x < 0 ㅠ if 0 < x < π on the interval [-π, π). Plot the graph of f(x) over the R and then compute an approximation of f(x) by using its Fourier expansion as in Equation (0.1) up to and including n = 4.
Suppose now that we have a periodic function f(x) with period 2 such that f(x) = { x + 1 if − π ≤ x < 0 ㅠ if 0 < x < π on the interval [-π, π). Plot the graph of f(x) over the R and then compute an approximation of f(x) by using its Fourier expansion as in Equation (0.1) up to and including n = 4.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Joseph Fourier (21 March 1768 - 16 May 1830) said that any con-
tinuous, periodic function f(x) with period 2π can be written as a
series:
(0.1)
where
f(x) = fo + Σ(an cos(nx) + bn sin(nx)
n>1
fo = 1/ ſ *_* ƒ(x) dx;
ㅠ -π
an
(f(x), cos(nx))
bn = (f(x), sin(nx)).
The constant term fo “centralizes" the function f so that
[
-π
=
(f(x) – fo) dx
- (1)
0
The numbers an and bn are called the Fourier coefficients of the func-
tion f(x). For small values of n, the functions cos(nx) and sin(nx)
capture to the low frequencies; when n is large, cos(nx) and sin(nx)
capture high frequencies. Thus we can use the Fourier expansion to
approximate periodic functions.
Suppose now that we have a periodic function f(x) with period 2
such that
f(x) = { 1 + 150;
+ 1 if − π ≤ x < 0
if 0 < x < π
on the interval [-π, π).
Plot the graph of f(x) over the R and then compute an approximation
of f(x) by using its Fourier expansion as in Equation (0.1) up to and
including n = 4.
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