Discrepancy in two-point correlator integral of Harmonic Oscillator

The two point correlator of Quantum H.O. of natural frequency $omega$, calculated using path integrals, is
$$C_{2}=Dleft(t_{2}-t_{1}right) propto int frac{d w^{prime}}{2 pi} frac{e^{-i w^{prime}left(t_{2}-t_{1}right)}}{w^{prime2}-w^{2}}tag{1}$$

which evaluated using contour integral in complex plane, comes out to be (can be verified by conventionally calculating it using ladder operators) –
$$C_{2}propto-ie^{-iw|t_2-t_1|}tag{2}$$

But $(1)$ is purely real as can be seen by decomposing exp. into sines and cosines –
$$ int frac{d w^{prime}}{2 pi} frac{cos(w^{prime}left(t_{2}-t_{1}right))}{w^{prime2}-w^{2}}-iint frac{d w^{prime}}{2 pi} frac{sin( w^{prime}left(t_{2}-t_{1}right))}{w^{prime2}-w^{2}}tag{3} $$

second term is an odd function therefore vanishes.

Why is there a discrepancy between $(1)$ and $(2)$?

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$bf{Edit1}$ : Another potential source of same doubt is Griffiths’ INTRODUCTION TO QUANTUM MECHANICS, first ed., eq. $bf{(11.50)}$ reads
$$G(mathbf{r})=frac{1}{(2 pi)^{2}} frac{2}{r} int_{0}^{infty} frac{s sin (s r)}{k^{2}-s^{2}} d s=frac{1}{4 pi^{2} r} int_{-infty}^{infty} frac{s sin (s r)}{k^{2}-s^{2}} d stag{11.50}$$
which is simplified version of $(1)$ in $3$D ($omega’$ renamed as $s$ and $omega$ as $k$). This is purely real.

He then evaluates it with contour integrals to (eq$bf{(11.55)}$) –
$$G(mathbf{r})=frac{i}{8 pi^{2} r}left[left(i pi e^{i k r}right)-left(-i pi e^{i k r}right)right]=-frac{e^{i k r}}{4 pi r}$$
which is not a real number.

Also, this may be helpful.

$bf{Edit2}$ : Doing the integral $(1)$ using two different contours –

$bf{1.}$With the following

for $t_2-t_1>0$ and the limit the radius of small semicircles goes to zero,
I got
$$C_2=frac{sin(omega x)}{2omega}tag{4}$$
(Note that the contributions from small semicircles do not vanish under the limit radius$rightarrow 0$)

$bf{2.}$Taking contour with shifted poles –

I got
$$C_2=-i frac{e^{-i omega x}}{2 omega}tag{5}$$

I know that using different contours to regulate the integral gives different results. But here’s what I’m thinking – It is obviously an improper integral hence we can only assign principle value to it. Also, both terms in $(3)$ diverges but second term’s integrand is an odd function. So, divergence in $omega’>0$ region would exactly cancel the divergence in $omega'<0$ region. There, $(1)$ is divergent(not "exactly" divergent, but is of the indeterminate form $infty$–$infty$) but still lies on real axis.

So I’m thinking, also inpired by comments, that deforming contour or shifting the poles would render the second term in $(3)$ not exactly an odd function. But consider the second contour, we shift $omega rightarrow omega-iepsilon$, carry out the integration and take limit $epsilon rightarrow 0^{+} $. In this limit, shouldn’t the result converge to a real number?

Also, Why we choose contour $bf{2}$ and not $bf{1}$ to calculate $bf{C_2}$ of H.O.?