Consider the period 4 function-2 < x < 00 < x < 2f(x + 4) otherwise.1f(x) = {2 - x((a) Sketch three periods of f(x).(b) Show that the Fourier coefficients b1, b2, ... may be written asb2m-1 = 01bamMTfor m = 1,2, ....(c) Knowing the Fourier coefficients4a2m = 0 , a2m-1 =(2m - 1)2п?for m = 1,2,..., write down the Fourier series corresponding to f(x).(d) Briefly explain why the Dirichlet conditions hold.(e) By evaluating f(x) at a suitable value of x, find1Σ(2m – 1)²m=1

Answered: Image /qna-images/answer/3bd5a6d6-9475-4a22-8b43-a516b198a5b5.jpg

Consider the period 4 function -2 < x < 0 0 < x < 2 1 f(x) = {2 - x ( f(x + 4) otherwise. (a) Sketch three periods of f(x). (b) Show that the Fourier coefficients b1, b2, ... may be written as b2m-1 = 0 1 bam for m = 1,2,....

Consider the period 4 function -2 < x < 0 0 < x < 2 1 f(x) = {2 - x ( f(x + 4) otherwise. (a) Sketch three periods of f(x). (b) Show that the Fourier coefficients b1, b2, ... may be written as b2m-1 = 0 1 bam for m = 1,2,....

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Consider the function f(x) = x + 2 sin(x) for - <x<. Find (f-1)'c0). (5-1)'(M) = help (numbers)

A: Note that for x=0, f(x)=0 Then we will find f'(0)

Q: 1 (-0 + 7) - 1 Sketch a graph of the function f(0) = -3 sin(-0+

Q: Let f(x) (x2 +x), -n<x<n be a periodic function, then the Fourier coefficient az %3D Answer:

A: The Fourier series for the function can be calculated using the formula fx=A0+∑n=1∞…

Q: then prove that i) F, [ƒ (x) cos ax] = [f, (p + a) +f, (p a)] 1 ii) F.[f(x) sin ax] 2 [f. (p + a) –…

Q: Sketch a graph of the function f(x) 2 sin(2x). %3D -77/8-31/4-5n/8-/2 -3/8-7/4-T/8 t/4 3n/8- /25/8…

A: Given function is f(x)=-2sin(2x)We know that period of sin(x)=2πThe period of sin(2x) will be π and…

Q: Find the Fourier coefficients of the periodic function f(x)=x-2x², (-<x<n); f(x+2) = f(x)

A: We have to solve by Fourier series.

Q: f(2) = cos 3z function , Find the maximum value in 0, 7] x [0, T]

Q: Find the Fourier approximation with the specified order of the function on the interval [0, 2π].f…

Q: (a) Show that the polynomial of degree 3 is differentiable in the complex plane, also find f'(z).…

A: For part (a): It is required to show that the polynomial of degree 3 is differentiable in the…

Q: The best approximation of f(x) = cos(8 + x) by a Fourier polynomial of degree 1 is

A: The Fourier Polynomial of degree n is Fn(x)=a0 + ∑k=1nakcos(kx) +∑k=1nbksin(kx). Then, the Fourier…

Q: prove that f(x) = sin*x + cos*x is a periodic function with periods

A: We use the Given Equations Prove that f(x) is a periodic function with period pie/2.

Q: Problem 1. (a) Prove that the function f(x) = x³ – 4x + 2 has three distinct roots in the interval…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Compute the coefficient Fourier an of the *-1, 0<r <a S(2) = { 0, *<I< 27

Q: (16) Show that: (1) The function f(z) is continuous in all complex plane. (2) (-1+i)-8(1+i).

Q: Problem 2. For the function defined by if -22x20 0<x<2 52 - 2 f(x) = 2x+2 sketch the 4- Period…

A: As per the question we are given a piecewise defined function f(x) with period 4 and we have to…

Q: Consider the period 4 function -2 <x < 0 0 <x < 2 f(r+4) otherwise. f(r) · 2 - x (a) Sketch three…

Q: Find the Fourier approximation f2 for if -1 < x < 1/1/201 2 otherwise. f(x) f₂(x) = 0 eBook = 1 X

Q: Prove that Fourier cosine transform of f(x) = cos(x), 0<x<3 is: 1/27 (³ sin 3(p-1), sin 3(p+1) p-1 +…

A: Introduction: A Fourier remodel (FT) is a mathematical remodel that decomposes features relying on…

Q: find Fourier coefficients

Q: 43 part a b and c

A: Topic:- function and it's Derivatives

Q: Find Fourierexpansion of the Punction fcx) = x2 XE [-T, TT] then deduce thetn = | 2' (572 (7)2t. (2)…

A: Since you have asked multiple questions, we will solve the first Question for you. If you want any…

Q: XP+2ap +2a') ple 23: Find the inverse Fourier sine transform f(x) of F, (P- ap

Q: f is a function with the following Fourier coefficients on the interval [-, π] ao= - a₁ = b₁ === a₂=…

Q: (x²–1)dx S (x++3x² +1)[tan-1 x2-1 aluation of is

A: Given integral is I=∫x2-1x4+3x2+1tan-1x2+1xdx To evaluate I Now, dividing the numerator and…

Q: ind and classify the Singularities of the function Z f(z) = @= (2-1) Sin the ZeroS -

Q: Q6) (a) Prove that Fourier cosine transform of f(x) = cos(x), 1 sin 3(p-1), sin 3(p+1) p-1 p+1 +…

Q: Sketch a graph of the function f(x) = 5 cos| 6+ 3 -87 -71 -6 -SA -47 -3T 27 -T 1 2 3n 47 57 6n 7n…

A: Given: The function

Q: The function (f) with a period of 2π is defined by: f(x) = cosh (x - 2π) if x∈[π ; 3π] a)…

Q: - use the Cauchy Residue Theorem to integrate the given function around the

Q: 8. Find the 6th order Fourier approximation of the function that is periodic and f(x) = x, -2<x <2

A: Given function is fx=x , -2≤x≤2 Compare the interval -2≤x≤2 with -L≤x≤L, we get L=2…

Q: 3) Let f(x) =) () Find the fourier Senies of fx nd skatch the the sum function on [-37,37). +++*…

A: we know Fourier series f(x)= 90+∑ n=1∞ancosnx+bnsin nx…

Question

Consider the period 4 function
-2 < x < 0
0 < x < 2
f(x + 4) otherwise.
1
f(x) = {2 - x
(
(a) Sketch three periods of f(x).
(b) Show that the Fourier coefficients b1, b2, ... may be written as
b2m-1 = 0
1
bam
MT
for m = 1,2, ....
(c) Knowing the Fourier coefficients
4
a2m = 0 , a2m-1 =
(2m - 1)2п?
for m = 1,2,..., write down the Fourier series corresponding to f(x).
(d) Briefly explain why the Dirichlet conditions hold.
(e) By evaluating f(x) at a suitable value of x, find
1
Σ
(2m – 1)²
m=1

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!

SEE SOLUTION Check out a sample Q&A here

Step 10

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Fourier Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Prove that the Fourier transform of f") (x), the nth derivative of f(x) is (- ip) times the Fourier transform of f(x) provided that the first (n – 1) derivatives of f(x) vanish as x→to.
Compute the derivatives
Graph and find the Fourier coefficients of the following functions f (x): - T/2 < x < Tn/2 1/2 < x < 3n/2 1 a) f(x) = { -1 b) f(x) = { "** -π
Compute the derivative of the given function. sin ¹(x³ + 4) O O O d -[sin ¹(x³ + 4)] = dx d dx d dx d dx √ + 1-(-³ 4) 2 (x³ + 4) [sin ¹(x³ + 4)] = √/1 + (x² + 4)² (3x+²) +4)2 [sin¯¹(x³ + 4)] = = -[sin¯¹(x^³ + 4)] = 3x² 1 - (x³ + 4)² 1 1 – (x-³. (3x-²) +4)²
جامعةرد sity of Nizwa is The period of the function f(x) = 3sin4(x-) TE 3 (a) (b) (b) (c) (4); 3 Select one: O (b)
prove that u=y3-3x²y is a harmonic function determine its harmonic conjugate and find the corresponding analytic function f(z)in the terms of z
(3) Compute an approximation of f(x) = tanx by computing the coefficients Co, C₁, C2, C3 (a = 0). Then graph y = tan x and the obtained cubic polynomial in the same coordinate system. As you zoom on the origin (0, 0) do these two graphs get closer?
sketch a graph of -4 sin(x - π) 6 5 4 3 2 1 -2π -3π/2 -J -Jt/2 -1 -2 -3 -4 -5 -6 π/2 B 3/2 2π
Let z be a complex number.
(1) 3. 3 Z = 6x, X2 + X, + X, + 106 find the endpcints of the function