we can express a fourier series in the form $$f(x) = \dfrac{1}{L} \sum_{k} e^{kix}\tilde{f_k}$$ where $L$ is the length of the interval of the fourier series and $k = \dfrac{2\pi n}{L}$ is the fourier-mode and $\tilde{f_k}$ the fourier component.
We resolve the fourier component with $$\tilde{f_k} = \int_Ie^{-inx}f(x)dx$$
Usually we do this for
- $k = 0$
- $k \neq 0$
1. Why do we consider these two different cases?
2. Are there other situations where we might consider more special cases?
Most times $k=0$ is problematic. Imagine the function $f(x)=1$ the integral would become $$\int e^{-inx}f(x)\mathrm{dx}=\int e^{-inx}\cdot 1\mathrm{dx}=\frac{e^{-inx}}{-in}+c$$
Obviously $n\neq 0$ must be given so that we can integrate the exponential. Hence, me must look at $n=0$ separately.
I dont tinkt to remember a case where i had to look at other cases, but I think it is possible. I am absolutely sure that you have to investigate different cases if you funcion $f(x)$ is strange (e.g. absolute value of something, etc.).