Question

Find the Fourier cosine series and sine series, respectively, for the even and odd periodic extensions of the following function: f(x)= x if 0<x<π/2.

2 if π/2<x<π.

Graph f with its periodic extensions (up to n = 4) using Mathematica.(leave codes here)

Answer #1

1. Find the Fourier cosine series for f(x) = x on the interval 0
≤ x ≤ π in terms of cos(kx). Hint: Use the even extension.
2. Find the Fourier sine series for f(x) = x on the interval 0 ≤
x ≤ 1 in terms of sin(kπx). Hint: Use the odd extension.

Find the:
(a) Fourier cosine series
(b) Fourier sine series
for the following shape using half range expressions
f(x)=x^(2), 0 less than or equal to x less than or equal to
1

Fourier Series
Expand each function into its cosine series and sine series for
the given period
P=2
f(x) = x, 0<=x<5
f(x) = 1, 5<=x<10

Fourier Series Expand each function into its cosine series and
sine series for the given period P = 2π f(x) = cos x

Expand the given function in an appropriate cosine or sine
series.
f(x) =
π,
−1 < x < 0
−π,
0 ≤ x < 1

f(x) = 2x - 7
x ∈ (0,7)
Draw a plot of the periodic Fourier Series expansion of f(x).
What is its value at x=0 and why? Is it odd or even?
Expand the given function in a Fourier Series also

a. Let f be an odd function. Find the Fourier series of f on
[-1, 1]
b. Let f be an even function. Find the Fourier series of f on
[-1, 1].
c. At what condition for f would make the series converge to f
at x=0 and x=1?

Write the Fourier cosine series for f(x) on the interval 0 ≤ x ≤
π. Parameter c is a constant. f(x) = x + e −x + c
(b) Determine the value of c such that a0 in the Fourier cosine
series is equal to zero.

Expand the following periodic functions in the interval –pi
< x < pi in a sine Cosine Fourier Series
1. F (x) = 0 -pi < x < pi/2
= 1 Pi/2 < x <pi
2. F(x) = 0 -pi <x < pi
= x 0 < x < pi

Find the half-range cosine Fourier series expansion of the
function f(x) = x + 3;
0 < x < 1.

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