In the proof of Poisson Summation Formula as presented in the text book Fourier Analysis:
$\begin{aligned} \int_{0}^{1}\left(\sum_{n=-\infty}^{\infty} f(x+n)\right) e^{-2 \pi i m x} d x &=\sum_{n=-\infty}^{\infty} \int_{0}^{1} f(x+n) e^{-2 \pi i m x} d x \\ &=\sum_{n=-\infty}^{\infty} \int_{n}^{n+1} f(y) e^{-2 \pi i m y} d y \\ &=\int_{-\infty}^{\infty} f(y) e^{-2 \pi i m y} d y \\ &=\hat{f}(m) \end{aligned}$
I understand a variable change of $y=x+n$ was made, but then how come $x$ in the power of $e$ became $y$ and not $(y-n)$.
$e^{-2\pi (y-k)}=e^{-2\pi iy}e^{-2\pi ik} = e^{-2\pi iy}$
$e^{-2\pi ik} = 1$ for integers $k$