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 ㅠ 1 ƒo = = = ["*_* f(x) dx; ㅠ -T an (f(x), cos(nx)) bn = (f(x), sin(nx)). The constant term fo “centralizes” the function f so that = [" (f(x) – fo) dx = 0 π 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 th we have a periodic function f(x) with period 2 such that f(x) = { ㅠ + 1 if -π ≤ x < 0 1- if 0 ≤ x < T on the interval [-T, 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. =
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 ㅠ 1 ƒo = = = ["*_* f(x) dx; ㅠ -T an (f(x), cos(nx)) bn = (f(x), sin(nx)). The constant term fo “centralizes” the function f so that = [" (f(x) – fo) dx = 0 π 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 th we have a periodic function f(x) with period 2 such that f(x) = { ㅠ + 1 if -π ≤ x < 0 1- if 0 ≤ x < T on the interval [-T, 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
Related questions
Question
Can you please do this without using Python.
Transcribed Image Text:(d) 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
ƒo = ²/7 ["*_* ƒ(x) dx;
ㅠ
-π
an (f(x), cos(n.x))
=
bn (f(x), sin(nx)).
The constant term fo "centralizes" the function f so that
[(f(x) – fo) dx = 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) =
={
π
+ 1 if − π ≤ x < 0
x
1- 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.
:
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
