- #1

NutriGrainKiller

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I understand what the Fourier Theorem means, as well as how it behaves, I just don't understand how the math actually pans out or in what order to do what.

I'm going to start off with what I know.

[tex]f(x) = \frac{a_0}{2} \sum_{n=1}^{\infty}(a_n}\cos{nx} + b_{n}\sin{nx})[/tex]

while,

[tex] a_{0} = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx[/tex]

[tex]a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos({nx})\ dx[/tex]

[tex]b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin({nx})\ dx[/tex]

This is of course only the case with periodic functions. Depending on how the graph looks it is possible to derive [tex]f(x)[/tex], the spatial period, and maybe even its tendency to be even or odd.

(Even/odd meaning whether or not the beginning of the wavelength is at the origin. If it is/does, it's odd and only contains cosine terms, if not and it behaves more like a sine wave (highest amplitude at origin) than it is even, thus not containing any cosine terms.)

if we are given [tex]f(x)[/tex], all we do is find [tex]a_{0}[/tex], [tex]a_{n}[/tex] and [tex] B_{n}[/tex] then plug into the first equation. Is this right? I am getting absurdly long answers doing this, and as far as I can tell I can't find any way of finding out whether I'm headed in the right direction or not.

Here is one of the problems I'm having trouble with:

[tex]f(x) = A\cos({\frac{\pi x}{\lambda}})[/tex], find the Fourier series (it is assumed the function is periodic on the interval [tex][0,2\lambda][/tex])

I'm going to start off with what I know.

[tex]f(x) = \frac{a_0}{2} \sum_{n=1}^{\infty}(a_n}\cos{nx} + b_{n}\sin{nx})[/tex]

while,

[tex] a_{0} = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx[/tex]

[tex]a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos({nx})\ dx[/tex]

[tex]b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin({nx})\ dx[/tex]

This is of course only the case with periodic functions. Depending on how the graph looks it is possible to derive [tex]f(x)[/tex], the spatial period, and maybe even its tendency to be even or odd.

(Even/odd meaning whether or not the beginning of the wavelength is at the origin. If it is/does, it's odd and only contains cosine terms, if not and it behaves more like a sine wave (highest amplitude at origin) than it is even, thus not containing any cosine terms.)

if we are given [tex]f(x)[/tex], all we do is find [tex]a_{0}[/tex], [tex]a_{n}[/tex] and [tex] B_{n}[/tex] then plug into the first equation. Is this right? I am getting absurdly long answers doing this, and as far as I can tell I can't find any way of finding out whether I'm headed in the right direction or not.

Here is one of the problems I'm having trouble with:

[tex]f(x) = A\cos({\frac{\pi x}{\lambda}})[/tex], find the Fourier series (it is assumed the function is periodic on the interval [tex][0,2\lambda][/tex])

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