Ms. Sarai Dickens V
Dec 18, 2024
Construct the first three Fourier approximations to the square wave function
Construct the first three Fourier approximations of the square wave function. Please provide clear details on the process involved, including any relevant mathematical formulas and explanations of the underlying concepts. Additionally, could you describe any assumptions made during the approximations and the significance of these approximations in practical applications?
1 Answers
Since f(x) is odd, it will be a pure sine series.
Then
Integral[-pi..pi] f(x)sin(nx) dx = Integral[0..pi] -2sin(nx) dx
= 2/n [ cos(n*pi) – cos(n*0) ]
= 2/n [ (-1)^n – 1 ]
which is
= 0 when n is even and
= -4/n when n is odd.
Since
Integral[-pi..pi] sin^2(nx) dx = pi,
we see that the Fourier series is
f(x) = -4/(1*pi) sin(x) – 4/(3*pi) sin(3x) – 4/(5*pi) sin(5x) – …
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