Question

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?

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)

Compute the complex Fourier series of the function f(x)= 0 if −
π < x < 0, 1 if 0 ≤ x < π
on the interval [−π, π]. To what value does the complex Fourier
series converge at x = 0?

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.

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

Find the Fourier series of the function f on the given
interval.
f(x) =
0,
−π < x < 0
1,
0 ≤ x < π

Find the Fourier series of the function:
f(x) =
{0, -pi < x < 0
{1, 0 <= x < pi

Find the fourier series representation of each periodic
function
f(x) = 0, -4 <x<0
f(x) = 8, 0<=x<=1
f(x) = 0, 1<x<4

Find the Fourier series for the following function (which has
period 2): f(x)= −x if −1<x<0
x if 0 < x < 1

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

Find the Fourier series of the half-range cosine
expansion (even) the function f(t) = c-t 0<t<c

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