Fourier series specific value

fourier analysis

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?

Best Answer

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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