Here is a common example in the discussion on orthonormal family. Let $\mathcal L = C[a, b]$. For $k \in \mathbb Z$, let $e\_k \in \mathcal L$ be defined by $$e\_k(\xi) := \frac{1}{\sqrt{b-a}} e^{\frac{2\pi i k(\xi - a)}{b-a}}, \forall \xi \in [a, b].$$ Then it is claimed that $\{e\_k\}\_{k=-\infty}^{+\infty}$ is an orthonormal family. It is clear to see that $$\langle e\_k, e\_l \rangle = \int\_a^b e\_k(\xi) \overline{e\_l(\xi)} d\xi = \frac{1}{b-a} \int\_a^b e^{\frac{2\pi i (k-l)(\xi - a)}{b-a}} d\xi.$$ From here I can see that the integral is $1$ when $k =l$. However, I could not figure out why it is $0$ when $k \neq l$. Basically, I could not do the integration. Could anyone explain how to integrate when $k \neq l$, please? Thank you!