Sines Fourier series

Question

There is a basic principle I don't get. Say I want to find the Sines Fourier series of $e^x$ on $[0,1]$. Why do I in this case treat a "continuation" of $e^x$ from $[0,1]$ to $[-1,1]$ so it is an odd function? I always see it being done but I don't know why. Of course the Fourier series should be odd, is it just a random minimal interval we choose to build the series on?

Answer 1

Using combination of sine functions, as you note it, we deal with an odd function.

You surely know that an odd function is such that

$$ \forall x, \ \ f(-x)=-f(x)$$

Therefore, the only odd function "continuating" $f(x)=e^x$ is $-f(-x)=-e^{-x}$ on $[-1,0]$ making it compulsory to consider this interval $[-1,0]$.

In a further step, we complete the function defined in this way on $[-1,1]$ by translating it with $2k\vec{i}$ translation vectors where $\vec{i}$ is the unit vector of $x$-axis.