Question

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

Answer #1

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

Expand the function f(x) = x^2 in a Fourier sine series on the
interval 0 ≤ x ≤ 1.

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

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)

(a) expand f(x)=8, 0<x<3 into cosine series period 6
(b) expand f(x)=8, 0<x<3 into a sine series period 6
(c) determine value each series converges to when x=42
(d) graph (b) for 3 periods, over the interval [-9,9]

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

Consider the first full period of the sine function:
sin(x), 0 < x < 2π.
(1) Plot the original function and your
four-term approximation using a computer for the range −2π < x
< 0. Comment.
(2) Expand sin(x), 0 < x < 2π, in a
Fourier sine series.

Find the half range cosine Fourier series expansion of the
function f(x) = x + 3, 0 < x < 1
Need full work shown (formulas/ every step)

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