I have trouble calculating below integral: $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{e^{i(\omega y+\beta x)}}{a+b\beta^2+c\omega^2+ik\beta}d\beta d\omega$$ where a, b, c, k are real constants and positive. $x$ and $y$ are also constant but not necessarily positive but they are real. I tried to follow Cauchy principal value based on my previous question in Integral containing exponential but I couldn't solve it. Please let me know if you need further information. Thanks and Happy Holiday.
Update 1: With some guidance, I followed below solution: $$\int_{-\infty}^{+\infty}e^{i\beta x}\int_{-\infty}^{+\infty}\frac{e^{i\omega y}}{a+b\beta^2+c\omega^2+ik\beta} d\omega d\beta$$ I focused at the internal integral (with respect to $\omega$). The denumonetor would have two roots: $$c\omega^2+(a+b\beta^2)+ik\beta$$ I used the post in here: How do I get the square root of a complex number? So the roots would be: $$\omega_r=\mp (\sqrt\frac{(a+b\beta^2)+\sqrt{(a+b\beta^2)^2+(k\beta)^2}}{2a}+i\sqrt\frac{-(a+b\beta^2)+\sqrt{(a+b\beta^2)^2+(k\beta)^2}}{2a})$$ Then I tried to follow complex integral considering positive roots is inside contour (please correct me if it is wrong). Doing so, I ended up with the internal integral to be equal to $\frac{\pi i}{\omega_r}e^{iy\omega_r}$. Considering this solution, the overall integral would become: $$\int_{-\infty}^{+\infty}\frac{\pi i}{\sqrt{A+B\beta^2+\sqrt{C\beta^4+D\beta^2+E}}+i\sqrt{-A-B\beta^2+\sqrt{C\beta^4+D\beta^2+E}}}e^{i(\beta x+(\sqrt{A+B\beta^2+\sqrt{C\beta^4+D\beta^2+E}}+i\sqrt{-A-B\beta^2+\sqrt{C\beta^4+D\beta^2+E}})y)}d\beta$$ At this stage, I don't think the denumenator has any root (??). I appreciate if you can take a look and confirm if I am in the right direction and how I can calculate the final integral. Thanks
Update 2: This time, I started from $\beta$ variable first. Doing so, I found the roots of the denominator as: $$\beta_r=\frac{-ik\mp\sqrt{-k^2-4ab-4bc\omega^2}}{-2b}$$ Note that in this equation, the parameters are positive so I write it as: $$\beta_r=\frac{-ik\mp i\sqrt{k^2+4ab+4bc\omega^2}}{-2b}$$ So based on this I found the integral becomes: $$-2\pi b\int_{-\infty}^{+\infty}\frac{e^{i\omega y-ikx+ ix\sqrt{k^2+4ab+4bc\omega^2}}}{\sqrt{k^2+4ab+4bc\omega^2}}d\omega$$ Then, The roots for the denominator would be: $$\omega_r= \mp i \sqrt(\frac{b}{c})$$. Now, I need to find residue at these roots?
