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Let $ f: \Bbb R \to \Bbb R $ be a function that satisfies the following property $$f(x+\pi)=-f(x),\ \text{for all $x\in\Bbb R$.}$$

Show that all Fourier coefficients of subscript pa are null.

Use this result to find the Fourier series of the function $ f (x) = x $ if $ 0 \le x \le \pi $, $ f (x + \pi) = - f (x) $ for all $ x \in \Bbb R $.

I already did the test that the even subscript coefficients are null. The question at this point, which I don't know if it will be wrongly stated, is that how do I apply this exercise to find the series if $ f (x) = x $ does not satisfy that $ f (x + \pi) = - f (x) $. Can someone help me with that particular piece?

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  • I think that they are defining the function as the identity on $[0,\pi]$, and then you extendend it to $\mathbb{R}$ using the relation $f(x+\pi)=-f(x)$. For example, $f(4)=f((4-\pi)+\pi)=-f((4-\pi))=\pi-4$. – Sebathon Aug 03 '21 at 02:57

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