## Poisson Summation Formula Proof

I'm having trouble understanding a couple of parts of the proof presented here in Knapp's Basic Real Analysis : Book link from Google (pgs 389-390).

He defines $c_n=\frac{1}{2L}\int_{-L}^{L}f(t)e^{-2\pi int/L}dt$. He then takes the Fourier series for L=1 on $F(x)=\sum_{n=-\infty}^{\infty}f(x+n)$ and gets $F(x)=\sum_{n=-\infty}^{\infty}e^{2\pi i nx}\int_{0}^{1}F(t)e^{-2\pi i nt}dt$ which seemingly identifies $c_n=\int_{0}^{1}F(t)e^{-2\pi i nt}dt$. This means to me that $F(t)e^{-2\pi int}$ is an even function of t, but this is not mentioned; am I missing something here?

My second question is when he seemingly substitutes $t+k\rightarrow t$ in the integral $\int_0^1 f(t+k)e^{-2\pi int}dt=\int_{k}^{k+1}f(t)e^{-2\pi int}dt.$ However shouldn't the exponential in the integrand be $e^{-2\pi in(t-k)}$?

Thank you for any help!