Expand in Fourier series:
Expand in fourier sine and fourier cosine series of: f(x) =
x(L-x), 0<x<L
Expand in fourier cosine series: f(x) = sinx, 0<x<pi
Expand in fourier series f(x) = 2pi*x-x^2, 0<x<2pi,
assuming that f is periodic of period 2pi, that is,
f(x+2pi)=f(x)
Use the Fourier transform to find the solution of the following
initial boundaryvalue Laplace equations
uxx + uyy = 0, −∞ < x < ∞ 0 < y < a,
u(x, 0) = f(x), u(x, a) = 0, −∞ < x < ∞
u(x, y) → 0 uniformlyiny as|x| → ∞.
4 a) Find the Fourier Integral representation of ?(?) = 1 ?? |?|
< 1
0 ?? |?| > 1
b) Find the Fourier Sine Transform of ?(?) = ? −|?| . Hence
evaluate ∫ ?????? 1+?2 ??.
Both the Fourier Series and the Discrete Fourier Transform are
calculated using summation. Explain the key differences in what the
inputs each of the Fourier Series and the DFT are AND the
requirements the inputs.