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I am learning Fourier series, and I must compute specific values for the Fourier series of the function $5x+7$ on the interval [-8,8].

The fourier series of a function on an interval [-L, L] is given by: $$ f(x)=\sum_{n=0}^{\infty} A_{n} \cos \left(\frac{n \pi x}{L}\right)+\sum_{n=1}^{\infty} B_{n} \sin \left(\frac{n \pi x}{L}\right) $$

  • I find first $A_0$: $$ A_{0}=\frac{1}{2 L} \int_{-L}^{L} 5x+7 d x $$ The first term disappears (odd), and integrating 7 on the interval gives $A_0 = 7$

  • Finding $A_m$ $$ A_{m}=\frac{1}{L} \int_{-L}^{L} (5x+7) \cos \left(\frac{m \pi x}{L}\right) d x \quad m=1,2,3, \ldots \\ = \int_{-L}^{L} 5x \cos \left(\frac{m \pi x}{L}\right) d x + \int_{-L}^{L} 7 \cos \left(\frac{m \pi x}{L}\right) d x $$ The first term is odd (odd * even = odd), the second is orthogonal, so $A_m=0$.

  • Finding $B_m$ in the same manner:

$$ B_{m}=\frac{1}{L} \int_{-L}^{L} (5x+7) \sin \left(\frac{m \pi x}{L}\right) d x \quad m=1,2,3, \ldots \\ = \int_{-L}^{L} 5x \sin \left(\frac{m \pi x}{L}\right) d x + \int_{-L}^{L} 7 \sin \left(\frac{m \pi x}{L}\right) d x $$ The second term is orthogonal, but the first is odd and can be integrated. By partial integration I find: $$B_{m}=\frac{-1^{n+1}}{n \pi}$$

ie my Fourier series is: $$7 + \sum_{n=1}^{\infty} \frac{-1^{n+1} 80}{n \pi} \sin \left(\frac{n \pi x}{8}\right)$$

EDIT: I asked about $f(0)$ and as @Ninad pointed in the comment, it was easy to calculate. I edited my question for clarity!

EDIT 2: added missing 80 in the fourier series.

My issue is when calculating special values for this series. The site I am following is not covering this part and focusing in computing Fourier series (if I am not blind!: https://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx) I cannot find a good example to follow (I don't understand how the person comes to their conclusion here: Computing value of fourier series)

So I would be grateful for an explanation on how one computes positive and negative values in the interval of a Fourier series, or some resources to read with exemplified process!

For example, how do I take f(2), and f(-2) and deal with the infinite sum?

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    $\begingroup$ $f(0)$ looks like $7$ on that graph, I'm not sure what you're confused about. $\endgroup$ Apr 23, 2021 at 8:10
  • $\begingroup$ Sorry, you are right! Looks like I cannot read the graph properly, I was reading the x-axis! But I am still confused about computing specific values. I edited my question, please have a look! $\endgroup$
    – Dovendyr
    Apr 23, 2021 at 8:53

1 Answer 1

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There is a number of useful results about the pointwise convergence of Fourier series.

Take $f$ to be integrable on $[-L, L]$ and have a Fourier series $\sum a_n \cos (n\pi x/L) + b_n \sin (n\pi x /L)$. Then two examples are

  1. If $f$ has left and right derivative at $x\in [-L, L]$ then the Fourier series converges at $x$ to $\tfrac{1}{2}(f(x+)+f(x-))$ where $f(x\pm)$ indicate the limits of $f(x+t)$ as $t\to x$ from above and below.
  2. If $f$ is continuous and the sum of the Fourier coefficients converges absolutely (i.e. $\sum |a_n| + |b_n|$ converges) then the Fourier series converges to $f(x)$.

Use example 1 applied to your case, (assuming the coefficients are OK) we see that $$ 17 = f(2) = 7 + \sum_{n=1}^\infty \frac{(-1)^{n+1}80}{n\pi} \sin \left( \tfrac{1}{4}n\pi \right) $$ where the infinite sum on the right will converge. You can calculate values $$ \sin \left( \tfrac{1}{4} n\pi \right)= \left\{ \begin{array}{2} \sqrt{2}/2&n=1,3,9,11, \cdots\\ 1&n=2,10, \cdots \\ -\sqrt{2}/2&n=5,7,13,15, \cdots \\ -1&n=6,14,\cdots \\ 0&n=4,8,12,\cdots \end{array} \right. $$ and if you combine into groups of eight terms you have, $$ \tfrac{1}{8}\pi = \sum_{n=0}^\infty \left\{ -\frac{\tfrac{1}{2}\sqrt{2}}{8n+1} + \frac{1}{8n+2} - \frac{\tfrac{1}{2}\sqrt{2}}{8n+3} + \frac{0}{8n+4} + \frac{\tfrac{1}{2}\sqrt{2}/2}{8n+5} -\frac{1}{8n+6}+\frac{\tfrac{1}{2}\sqrt{2}/2}{8n+7}-\frac{0}{8n+8} \right\}$$

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  • $\begingroup$ Thank you very much @WA Don, that was very informative! So the important is to generate a series of terms that are repeating (cyclic?) $\endgroup$
    – Dovendyr
    Apr 23, 2021 at 16:52
  • $\begingroup$ No - that won't always happen as it did in this case. The important thing is that the Fourier series converges for the right type of $f$ so you can then say the $f(x)$ and the sum of the series are the same. For example, to find $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ you can develop a Fourier series for a simple function (Maybe $f(x) = |x|$ on $[-\pi,\pi]?)$ which gives the $1/n^2$ terms when $x=0$ and obtain the famous $$\sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ $\endgroup$
    – WA Don
    Apr 24, 2021 at 8:40
  • $\begingroup$ I will file this knowledge for later, we just started with Fourier and did not yet come ton convergence :) $\endgroup$
    – Dovendyr
    Apr 24, 2021 at 8:47
  • $\begingroup$ Btw sorry there was a typo in my question, 80 is missing in B_m, let me fix that... $\endgroup$
    – Dovendyr
    Apr 24, 2021 at 8:47
  • $\begingroup$ By the way I have a follow up question if you have time and energy :) math.stackexchange.com/questions/4114511/… $\endgroup$
    – Dovendyr
    Apr 24, 2021 at 8:50

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