The Fourier series (complex form) for continuous function $f(x)$ defined on $[-l, l]$ is $$ \sum_{k = -\infty}^{\infty} c_{k} e^{ik \pi x/l} $$
where $$ c_{k} = \frac{1}{2l} \int_{-1}^{l} f(x) e^{-ik\pi x/l}dx $$
Now if I have a continuous function $g(x)$ defined on $[0, 2\pi]$, then I can shift the function to the interval $[-\pi, \pi]$ by defining a new function $$ f(x) = g(x+\pi), \:\: x \in [-\pi, \pi]$$
So in terms of Fourier series, the coefficient $c_{k}$ is therefore
$$ c_{k} = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-ik\pi x/\pi} dx $$
$$ = \frac{1}{2\pi} \int_{-\pi}^{\pi} g(x+\pi) e^{-ik\pi x/\pi} dx $$
Now let $y = x+\pi$, then the above is
$$ c_{k} = \frac{1}{2\pi} \int_{0}^{2\pi} g(y) e^{-ik\pi (y-\pi)/\pi} dy $$
or $$ c_{k} = \frac{1}{2\pi} \int_{0}^{2\pi} g(y) e^{-iky} e^{i k \pi} dy $$
So we have that
$$ c_{k} = \begin{cases} \frac{1}{2\pi} \int_{0}^{2\pi} g(y) e^{-iky}dy, \:\:\:\: k \:\: \text{ even} \\ - \frac{1}{2\pi} \int_{0}^{2\pi} g(y) e^{-iky} dy, \:\:\:\: k \:\: \text{ odd} \end{cases} $$
Problem
But according to a paper here: https://researchspace.ukzn.ac.za/bitstream/handle/10413/11809/Sibandze_Dan_Behlule_2013.pdf;sequence=1 (see page 4 equation $1.1$) ....
.... the coefficient $c_{k}$ for a function $g(x)$ defined on $[0,2\pi]$ is computed as
$$ c_{k} = \frac{1}{2\pi} \int_{0}^{2\pi} g(x) e^{-ikx}dx, $$
do I miss something here?