My question is about how to Fourier transform with respect to a variable t, the following function
$$\hat{P}(t,k)= \frac{\partial}{\partial t} \displaystyle\int_{t}^{+\infty} P_0(|t_1|,k)J_0\Bigg( \frac{kh_1}{t_1}\sqrt{t_1^2 - t^2}\Bigg) dt_1$$
where $h_1$ is a constant, $J_0$ is the Bessel function of the first kind and order zero. The author of the paper that i was reading(Theory of differential offset continuation; Sergey Fomel,2003), said that the Fourier transform of $\hat{P}(t,k)$ with respect to t was a known integral, and could be expressed as
$$\hat{P}(\omega,k)= \displaystyle i\int_{-\infty}^{+\infty} P_0(|t_1|,k)\frac{\sin(\omega|t_1|A)}{A} \,\,\,dt_1$$
where
$$\displaystyle A=\sqrt{1 + \Bigg(\frac{kh_1}{\omega t_1}\Bigg)^2}$$
Someone could please help me to figure out, how that is done?