Question

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

Answer #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)

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

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)

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.

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

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.

a) Find the Fourier cosine transform of e^(-ax), given
a>0.
b) Use item (a) above to find the Fourier sine transform of
e^(-ax)/x, given a > 0.

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

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

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