I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t\_+^{1/2}$? (Here $t\_+=\max (t,0)$.)



Fourier transform of tempered distribution
$$\tfrac{1}{2}\int\_{0}^{\infty}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)+\tfrac{1}{2}i\omega^{-1}$$


$$\int\_{0}^{\infty}te^{i\omega t}dt=-i\frac{d}{d\omega}\int\_{0}^{\infty}e^{i\omega t}dt=-i\pi\frac{d}{d\omega}\delta(\omega)-\omega^{-2}$$


$$\int\_{0}^{\infty}[\sqrt{t(1+t)}-t-\tfrac{1}{2}]e^{i\omega t}dt=\omega^{-2}-\tfrac{1}{2}i\omega^{-1}+\tfrac{1}{2}i\omega^{-1}e^{-i\omega/2}K\_{1}(-i\omega/2)
$$
adding these three results gives the required Fourier transform
$$
\int\_{0}^{\infty}\sqrt{t(1+t)}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)-i\pi\frac{d}{d\omega}\delta(\omega)+\tfrac{1}{2}i\omega^{-1}e^{-i\omega/2}K\_{1}(-i\omega/2)
$$
with $K\_1$ a modified Bessel function of the second kind; the derivative of the Dirac delta function should be understood in the context of an integral,
$$\int\_{-\infty}^{\infty}f(\omega)\frac{d}{d\omega}\delta(\omega)d\omega=-\lim\_{\omega\rightarrow 0}\frac{d}{d\omega}f(\omega)$$


