Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$
(1) Calculate the coefficients of the Full Fourier Series of $\phi$ over the interval $[-l,l]$ under their complex form
(2) Deduce from this result the expression of the coefficients of the Full Fourier series of $\phi$ over $[-l,l]$ under their real form.
Here is what I have so far:
(1) Note $\phi(x)={1\over{2l}}\sum_{n=-\infty}^\infty c_ne^{{in\pi x}\over{l}}$ $$c_n={1\over{2l}}\int_{-l}^le^xe^{{-in\pi x}\over l}dx = {1\over{2l}}\left[{1\over{1-{in\pi\over{l}}}}e^{x\left(1-{in\pi\over{l}}\right)}\right]_{-l}^l={1\over{2l}}\left[{l\over{l-in\pi}}\left(e^{l-in\pi}-e^{-l+in\pi}\right)\right] $$
Evaluating when n is odd and n is even:$c_n={(-1)^n\over{2l}}\left[{l\over{l-in\pi}}sinh(l)\right]$
$$\phi(x)=\sum_{n=-\infty}^{\infty}{(-1)^n\over{2l}}\left[{l\over{l-in\pi}}sinh(l)\right]e^{{in\pi x}\over{l}} $$
Now, I'm not sure how to use this to determine the coefficients of the full Fourier series. Any help would be appreciated
What you have so far is the Fourier series in complex form. To turn it into real form, plug $$e^{{in\pi x}\over{l}} = \cos {n\pi x \over l} + i \sin {n\pi x \over l}$$ into the series. After multiplying this by $$\left[{l\over{l-in\pi}}\sinh(l)\right]$$ separate the real and imaginary parts. The imaginary part is going to add up to $0$ since the function is real (indeed, the term labeled $-n$ kills the term labeled $n$), so you may as well drop it now. The real part is the series you want.