Calculating the Fourier Transform of 1/(a^2 - x^2): Seeking Guidance
Hello fellow mathematicians,
I am currently working on a problem involving the calculation of the Fourier transform of a particular function, and I could use some guidance to proceed. The function in question is given by:
$$\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \frac{e^{i x t}}{a^{2}-x^{2}} dx$$
where a is a constant and we need to consider two cases: t>0 and t<0.
I've made some progress in simplifying the integral as follows:
$$=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \frac{1}{2 a}\left(\frac{1}{a-x}+\frac{1}{a+x}\right) e^{i x t} dx \\ =\frac{1}{4 \pi a}\left[\int_{-\infty}^{+\infty} \frac{e^{i x t}}{a-x} dx+\int_{-\infty}^{+\infty} \frac{e^{i x t}}{a+x} dx\right].$$
Now, I'm looking to further simplify these integrals and evaluate them using the properties of the Fourier transform. Could anyone provide insight into how to proceed from this point? Any help, suggestions, or references to relevant techniques would be greatly appreciated.
