Question

In the interval −π < t < 0, f(t) = 1; and for 0 < t < π, f(t) = 0. f(t) = f(t+2 π)

Find the following for f(t) as associated with the Fourier series:

- a
_{0}=? - a
_{n}=? - b
_{n}=?

ω_{o} =?

Answer #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.

The sketch of the following periodic function f(t)
given in one period,
f(t) = {(3t+1), -1 < t <= 1 and
0, -3 < t <= -1
a) Find period of the function, 2p?
b) Find Fourier coeff, a0, an (n
=>1), bn?
c) Fourier series representation of f(t)?
d) Result from (c), find the
first four non-zero term?

Find the Fourier analysis (a0, an, bn)
0 < x < T/2 f(t)= 2At/T
T/2 < x < T f(t)= 2At/T - 2A

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

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.

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?

Calculate the Fourier series expansion of the function:
f(x)
=1/2(π-x) , when 0
< x ≤ π and
f(x) = -
1/2(π+x), when -π
≤ x < 0

For the periodic function y(t) with period 12, y(t) = (0 if −6
< t < −3, 4 if −3 < t < 3, 0 if 3 < t < 6)
(a) determine the real Fourier series of y(t): y(t) = a0 + ∞ Sum
n=1 (an cos(2πfnt) + bn sin (2πfnt))
The ns are subscripts

Solve the following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

Deduce Fourier series of sin x, over the interval : 0
< x < π

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