It is well known that the propagator (kernel) of a simple harmonic oscillator is given by $$ U\left(x_{b},T;x_{a},0\right)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{ \frac{im\omega}{2\hbar\sin\omega T}\left[\left(x_{a}^{2}+x_{b}^{2}\right)\cos\omega T-2x_{a}x_{b}\right]\right\}. \tag{1} $$
I want to show explicitly that at times that are integer multiples of the period (i.e. $T=(2\pi/\omega) n$) the propagator becomes $\delta (x_b - x_a)$ while for odd multiples of the period (i.e. $T=(\pi/\omega)(2n+1)$) it's equal to $\delta(x_b + x_a)$.
Proving the first case seems straightforward. By defining
$$ \epsilon=\sqrt{-\frac{\hbar\sin \omega T}{im\omega}} $$ we can rewrite the propagator as $$ U\left(x_{b},T;x_{a},0\right)=\frac{1}{\epsilon\sqrt{2\pi}}\exp\left\{ -\frac{1}{2\epsilon^{2}}\left[\left(x_{a}^{2}+x_{b}^{2}\right)\cos\omega T-2x_{a}x_{b}\right]\right\}. $$
Now, for $T\to2\pi n/\omega$ where $n$ is an integer, the propagator coincides with the definition of Dirac delta as a limit of a Gaussian:
$$ U\left(x_{b},T;x_{a},0\right)=\lim_{\epsilon\to0}\frac{1}{\epsilon\sqrt{2\pi}}\exp\left[-\frac{1}{2\epsilon^{2}}\left(x_{a}-x_{b}\right)^{2}\right]=\delta\left(x_{a}-x_{b}\right). $$ So far, so good. However, when $T\to\frac{\pi}{\omega}\left(2n+1\right)$, we still have $\epsilon \to 0$ except now $\cos \omega T \to -1$ and thus
$$ U\left(x_{b},T;x_{a},0\right)=\lim_{\epsilon\to0}\frac{1}{\epsilon\sqrt{2\pi}}\exp\left\{ \frac{1}{2\epsilon^{2}}\left(x_{a}+x_{b}\right)^{2}\right\}. $$
But now the exponent doesn't coincide with $\delta(x_a + x_b)$. Redefining $\epsilon$ such that it would behave appropriately inside the exponent leads to an overall imaginary phase, which isn't good either. What am I missing?
