Let $c_{L},c_{T},\omega$ be positive constants with $c_{L}>c_{T}$. Define
$p=\sqrt{\frac{\omega^{2}}{c_{L}^{2}}-\xi^{2}}\qquad q=\sqrt{\frac{\omega^{2}}{c_{T}^{2}}-\xi^{2}}$
Consider the function $D_{S}\left(\xi\right)$ defined as follows:
$D_{S}=4\xi^{2}pq\sin p\cos q+\left(\xi^{2}-q^{2}\right)^{2}\cos p\sin q$
The function $D_{S}(\xi)$ has some zeros in the real axis. I need to know if all the zeros are on the real axis. My question is: Is it true that $D_{S}(\xi)$ has no zeros on the upper-half of the complex plane? I.e., that $D_{S}(\xi)\neq0$ whenever $\text{Im }\xi>0$? Thanks.