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 $t_+=\max (t,0)$.)
1 Answer
$$\tfrac{1}{2}\int_{0}^{\infty}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)+\tfrac{1}{2}i\omega^{1}$$
$$\int_{0}^{\infty}te^{i\omega t}dt=i\frac{d}{d\omega}\int_{0}^{\infty}e^{i\omega t}dt=i\pi\frac{d}{d\omega}\delta(\omega)\omega^{2}$$
$$\int_{0}^{\infty}[\sqrt{t(1+t)}t\tfrac{1}{2}]e^{i\omega t}dt=\omega^{2}\tfrac{1}{2}i\omega^{1}+\tfrac{1}{2}i\omega^{1}e^{i\omega/2}K_{1}(i\omega/2) $$ 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, $$\int_{\infty}^{\infty}f(\omega)\frac{d}{d\omega}\delta(\omega)d\omega=\lim_{\omega\rightarrow 0}\frac{d}{d\omega}f(\omega)$$

1$\begingroup$ what's your reference for the second integral, the one with the Bessel function? $\endgroup$ Jul 3, 2013 at 14:21

1$\begingroup$ The second integral is the output of Mathematica, with the condition ${\rm Im}\,\omega>0$. Since the integrand decays as $1/t$ for large $t$, this condition can be extended to ${\rm Im}\,\omega\geq 0$. $\endgroup$ Jul 3, 2013 at 14:42

1$\begingroup$ Alternatively, it can be justified by referring to one of the integral representations for $K_1$. Like this one: dlmf.nist.gov/10.32.E8 $\endgroup$ Jul 3, 2013 at 14:51

1$\begingroup$ @Alex A: $\int_0^{\infty}t^{1}\sin(\omega t)dt=\pi/2\times{\rm sign}(\omega)$ for $\omega$ real; similarly, the integral that decays as $t^{1}\cos\omega t$ is finite. OK? $\endgroup$ Jul 12, 2013 at 15:04

1$\begingroup$ @Alex A: you are right, my mistake, I have corrected it; the final answer now contains an extra term $(\pi/2)\delta(\omega)$ $\endgroup$ Jul 24, 2013 at 10:52