Figure 1 shows a periodic function, f(x), with period of 2n. f(x) mim –π 0 π ao = 2A T (a) State if the function f(x) has odd or even symmetry. (b) Show that the Fourier coefficients of f(x) are given by: N a₁ = Figure 1 f(x) = X 2A 1+(-1)" 1-n² T T f(x)=< 2A 44(1 A is a constant. Hint: You will find the result in question 1 (b) useful to find an. Also, remember that a ƒ ƒodd (X)dx=0, ƒ ƒven(x)dx = 2] ƒeven (x)dx -a -a 0 b₁ = 0 (c) By evaluating the first six Fourier coefficients show that the Fourier series of f(x) is given by: Asin x if 0≤x≤n -Asinx if -≤x≤0 1 1 44 ( cos(2x) + = cos(4x) + = cos( TU 3 15 35 -cos(6x)+...
Figure 1 shows a periodic function, f(x), with period of 2n. f(x) mim –π 0 π ao = 2A T (a) State if the function f(x) has odd or even symmetry. (b) Show that the Fourier coefficients of f(x) are given by: N a₁ = Figure 1 f(x) = X 2A 1+(-1)" 1-n² T T f(x)=< 2A 44(1 A is a constant. Hint: You will find the result in question 1 (b) useful to find an. Also, remember that a ƒ ƒodd (X)dx=0, ƒ ƒven(x)dx = 2] ƒeven (x)dx -a -a 0 b₁ = 0 (c) By evaluating the first six Fourier coefficients show that the Fourier series of f(x) is given by: Asin x if 0≤x≤n -Asinx if -≤x≤0 1 1 44 ( cos(2x) + = cos(4x) + = cos( TU 3 15 35 -cos(6x)+...
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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Please explain all the steps
Transcribed Image Text:Figure 1 shows a periodic function, f(x), with period of 2π.
f(x)
wimm
-π O
π
ao =
(a) State if the function f(x) has odd or even symmetry.
2 A
(b) Show that the Fourier coefficients of f(x) are given by:
TC
Figure 1
a₁ =
2A 1+(-1)"
1-n²
T
f(x)= Asinx if 0≤x<a
- Asinx if -≤x≤0
f(x)=
A is a constant.
b₁ = 0
Hint: You will find the result in question 1(b) useful to find an. Also, remember
that
ƒ ƒodt (x) dx = 0, ƒ ƒeven (x) dx = 2 feven (x)dx
-a
-a
0
- 24_44 ( cos(2x) + = cos(4x) +
2A
TU
TU
15
(c) By evaluating the first six Fourier coefficients show that the Fourier series of f(x)
is given by:
1
+cos
35
-cos(6x)+...
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it should be (1+n)(1-n) for part b, you did (n+1)(n-1), please correct this
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