# Locating the zeros of a function

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.

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Adding some motivation would turn this seemingly random question into something that might interest someone :) –  Mariano Suárez-Alvarez Oct 3 '11 at 4:51
What are "lamb-waves"? (as per the tag) –  Gerry Myerson Oct 3 '11 at 5:49
@Gerry I'm killing the lamb-waves tag. That is way too specific a tag to be useful. It refers to a physical phenomenon that has rather little to do with mathematics. The only reason that I can see that the question is tagged so is that expressions in the question came from study of mathematical physics. –  Willie Wong Oct 3 '11 at 8:04
Assuming you're using the principal branch of the square root, the answer appears to be that there can be many such zeros. Take $\omega = c_L = 1$, $c_T = 1/2$. Then, for example, there is a zero at approximately $\xi = 1.129125082+1.323674424 i$.