For a function $f(x)$ on $\mathbb{R}$, its Fourier Transform is given by
$$F(\xi) = \frac{1}{2\pi}\int_{R} f(x)e^{-i \xi x}dx $$.
Then, what is the Fourier Transform of the wave function $f(x) = e^{i \xi x} $ itself? If I simply plug it in, I would get $\infty$. What does this mean?
Also, if $f(x) = e^{i \eta x}$, where $\eta \not = \xi$, what is its Fourier Transform?