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i'm studying Fourier series by myself and all the books beguin with. "Let f be a piecewise continuous function."

1.Why does the function has to be piecewise continuous?

Also, they define $f$ on a symetric and closed interval, for example, $[-\pi,\pi]$ but in some examples they do not consider them. $f(x)=x, -\pi\leq x \leq 0$ and $f(x)=0, 0

2.What happens with the value of $f$ on $\pi$?

About Fourier Transform.

  1. Why do they go to complex numbers to define the Fourier Transform?

  2. Another application of Fourier Transform, more than solutions to pde?

Thank you very much

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    We don't care of $f(\pi)$ when looking at $\int_{-\pi}^\pi f(t) e^{-int}dt$. We use the complex numbers because the complex exponentials $\{e^{int}\}_{n \in \mathbb{Z}}$ is an orthogonal basis of... the vector space of periodic functions. Usually, the book begin with "let $f$ be $C^1$ and $2\pi$ periodic" because in this case it is easier to show that $f$ is equal to its Fourier series. – 2017-01-17
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    Piecewise continuity already introduces enough "excitement" for an introduction to the Fourier transform. – 2017-01-17
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    Thank you very much @user1952009 for the detalis. – 2017-01-17
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    Dear @hardmath do not understand what you mean, could you explain me please? – 2017-01-17
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    I mean that while this is not the most general setting in which you can define the Fourier transform, it is general enough to be able to prove many useful results and to get many interesting examples/counterexamples. – 2017-01-17

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  1. The functions don't have to be piecewise continuous. You're probably dealing with a text at the level of Riemann integration. In that case, you could just assume the function is Riemann integrable on $[-\pi,\pi]$. Piecewise continuous is a common context for Riemann integration.

  2. The coefficients for the Fourier series are defined by integrals of the function. So, changing the function at any finite number of points is not going to alter the Fourier coefficients because the coefficients are integrals of $f$. What is affected is whether or not the resulting series will converge to $f$ as a particular point; if the Fourier series for $f$ converges to $f(x)$ at some $x$, then this will no longer hold if you change the value of $f$ at $x$, because the Fourier series doesn't change.

  3. Complex exponentials are much more convenient. But you could study the Fourier series using only the real trigonometric functions $$ f \sim \frac{1}{2}a_0 + a_1\cos(x)+b_1\sin(x)+a_2\cos(2x)+b_2\sin(2x)+\cdots. $$ This form is more cumbersome because (a) of the pesky $\frac{1}{2}$ multiplying $a_0$ and (b) because of having to deal with cosine and sine terms instead. The original Fourier series posed by Fourier used real functions. The exponential form is notationally much simpler, but equivalent.

  4. Signal analysis. MP4, MP3, JPEG file formats where discrete Fourier Analysis is used to approximate discrete data, and then filtering techniques are used to get rid of components beyond human perception, and where compression is then applied to the Fourier coefficients to reduce the file/data stream sizes.