- #1
Jncik
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Homework Statement
I have a function
f(x) = x^2/4 for |x|<π
I have the Fourier series of this function which is
and I need to prove that
The Attempt at a Solution
I tried to use dirichlet for x = 0 but I get -pi^2/3
Jncik said:actually for x = 0 I said that since we want 1/n^2
instead of (-1)^n
I will have to use (-1)^2n
hence I will get a 1/4 in the right side of the equation and a -pi^2/12 in the left side
hence -4pi^2/12 will be the answer..
how did you find yours?
as for x = pi
indeed, it's not continuous in this point hence
I will have
(pi^2/4 + 0)/2 = pi^2/8
pi^2/8 = pi^2/12 + 1/4 * x => x = pi^2/6
indeed it's correct...
but can you please explain me how you found your answer?
thanks in advance
Jncik said:i think that we assume that it's periodic... but I'm not sure...
yes I was wrong
but if I put x = pi and cancel them out won't the result be -pi^2/12?
isn't this wrong?
A Fourier expansion is a mathematical technique used to represent a periodic function as a sum of sine and cosine functions with different frequencies and amplitudes.
To find the summation of a function using Fourier expansion, you need to determine the coefficients of the sine and cosine functions in the expansion. These coefficients can be calculated using integrals and the properties of the Fourier series.
Finding the summation of a function using Fourier expansion can help us better understand the behavior of the function and make predictions about its values at different points. It is also a useful tool in solving differential equations and other mathematical problems.
No, not every function can be represented accurately using a Fourier expansion. The function must be periodic and have a finite number of discontinuities for the Fourier series to accurately represent it.
Fourier expansion is used in various scientific fields such as signal processing, image and sound compression, and solving differential equations in physics and engineering. It is also used in analyzing and predicting the behavior of waves in different systems.