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Math Help - Fourier Series - A few questions

  1. #1
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    Fourier Series - A few questions

    Hi, I have a few questions about Fourier series.

    I understand the derivation of the formulas for working out the coefficients in your series, but I have a few queries about your bounds of integration, and whether or not you need to change these bounds depending on what domain you are working in

    For example, when I calculate the fourier series for f(x) = 2x^2 over the domain of -1\leq x < 0 and f(x+1) = f(x), \forall x \in \mathbb{R}.... I get the same as if my function was defined as f(x) = 2x^2 when x \in [0,1). And then f(x) = f(x+1),\forall x \in \mathbb{R} I believe these functions do NOT look the same.

    Avoid my abuse of notation, but I only want to talk about bounds of integration....
    When integrating we get \int_{-L}^{L}=\int_{-\frac{1}{2}}^{\frac{1}{2}}.

    Now my professor has said that since the function is periodic we can 'shift' the bounds to make it easier, for example change it to \int_0^{1}.

    The thing I'm having difficulty with is that these integrals do not evaluate to the same thing. Take the example of f(x)=2x^2. Now drawing the function over the x-axis, I can see that the area under the curve(s) from -1/2 to 1/2 is the same as the area under the curve from 0 to 1. So I can see the motivation there. Am I stuffing up here because in the case when you consider in the domain of x\in [-1,0) that when you integrate from 0 to 1, if you look at the graph, it's actually the function 2(1-x)^2? Since you have shifted over, but you are still just integrating 2x^2 which may cause some trouble...

    If this isn't it... i'm really stuck... because I end up getting the same Fourier series when the function is initially defined in 2 different domains. And these functions do NOT look the same!

    Any help would be appreciated, thanks!
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  2. #2
    Grand Panjandrum
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    Re: Fourier Series - A few questions

    Quote Originally Posted by mathswannabe View Post
    Hi, I have a few questions about Fourier series.

    I understand the derivation of the formulas for working out the coefficients in your series, but I have a few queries about your bounds of integration, and whether or not you need to change these bounds depending on what domain you are working in

    For example, when I calculate the fourier series for f(x) = 2x^2 over the domain of -1\leq x < 0 and f(x+1) = f(x), \forall x \in \mathbb{R}.... I get the same as if my function was defined as f(x) = 2x^2 when x \in [0,1). And then f(x) = f(x+1),\forall x \in \mathbb{R} I believe these functions do NOT look the same.

    Avoid my abuse of notation, but I only want to talk about bounds of integration....
    When integrating we get \int_{-L}^{L}=\int_{-\frac{1}{2}}^{\frac{1}{2}}.

    Now my professor has said that since the function is periodic we can 'shift' the bounds to make it easier, for example change it to \int_0^{1}.

    The thing I'm having difficulty with is that these integrals do not evaluate to the same thing. Take the example of f(x)=2x^2. Now drawing the function over the x-axis, I can see that the area under the curve(s) from -1/2 to 1/2 is the same as the area under the curve from 0 to 1. So I can see the motivation there. Am I stuffing up here because in the case when you consider in the domain of x\in [-1,0) that when you integrate from 0 to 1, if you look at the graph, it's actually the function 2(1-x)^2? Since you have shifted over, but you are still just integrating 2x^2 which may cause some trouble...

    If this isn't it... i'm really stuck... because I end up getting the same Fourier series when the function is initially defined in 2 different domains. And these functions do NOT look the same!

    Any help would be appreciated, thanks!
    Your function is periodic so you can use any bounds that correspond to a periiod. But you have to use the right function values.

    So:

    f(x)=2x^2,\ \ x\in [-1,0) and periodic with period 1

    is:

    f(x)=2(x-1)^2,\ \ x \in [0,1) and periodic with period 1

    CB
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