Show that $ \int_{0}^{\frac{1}{f}} e^{-j2\pi kft} \cdot e^{j2\pi lft} dt=0$ for $k,l \in \mathbb{Z}$ with $k \ne l$

Question

How can I show that: $$\int\limits_{0}^{1/f} e^{-j2\pi kft} \cdot e^{j2\pi lft}dt=0 \qquad\forall(k,l) \in \mathbb{Z},\,k \ne l$$ I am not quite sure how to get from LHS to RHS. Can someone explain me the steps inbetween?

Answer 1

using what @Benjamin Wang mentioned, for the function you get: $$f(t)=\exp\left[2\pi fj t(l-k)\right]\qquad t\in[0,1/f]$$ now since $(l-k)\in\mathbb{Z}$ we have an integer multiple of a period happening over the integral, and the integral over a single period is zero, and so the integral must be zero. The exception being $l=k$ where $f(t)=1\forall t$ and so we negate this