The scalar propagator for the Klein-Gordon Lagrangian is given by:
$$D(x-y)=\int \frac{d^{4} k}{(2 \pi)^{4}} \frac{e^{i k(x-y)}}{k^{2}-m^{2}+i \varepsilon}$$
I need to know if it is an even function, i.e.:
$$D(x-y) = D(-(x-y)) = \int \frac{d^{4} k}{(2 \pi)^{4}} \frac{e^{i k(x-y)}}{k^{2}-m^{2}+i \varepsilon}\tag{1}$$
Since $k^2=k_0^2-\vec{k}^2$ and $k_0^2$ and $\vec{k}^2$ are both positive.
Also
$$\int_{-\infty}^{\infty} d^4k e^{ik(x-y)}=\int_{-\infty}^{\infty} d^4k e^{-ik(x-y)}=\int_{-\infty}^{\infty} d^4k e^{ik(y-x)}$$
So can we conclude that equation 1 is indeed true?
edit: rephrased the question hope it makes sense now