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If we have a standard function, like $f(x) = x$ or $g(x) = x^2$, defined between $0$ and $\pi$, then why should we be interested in extending this function to give a Fourier series that resembles this function between $0$ and $\pi$?

What is the whole purpose of this process?

Does it have any real life application or is it just a mathematical exercise?

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    $\begingroup$ Yes, it has applications. In physics, engineering, and even in mathematics (such as PDEs). $\endgroup$ – Gerald Edgar Mar 7 '15 at 21:49

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