What is Fourier transform of unilateral sinc function? $$\int_{0}^{+\infty} \frac{\sin(x)}{x} e^{itx} dx = ?$$ 
I know the Laplace transform of sinc is $\arctan(1/t)$. However, what if $t$ is a complex number?

In term of distributions, the Fourier transform of $\frac{\sin(\pi x)}{\pi x}$ is $1_{|\xi| < 1/2}$ and the FT of $1_{x > 0}$ is $\frac{1}{2i \pi}\frac{d}{d\xi} \log |\xi| + \frac{1}{2} \delta(\xi) $,
thus the FT of $\frac{\sin(\pi x)}{\pi x}1_{x > 0}$ is $$1_{|\xi| < 1/2} \ast (\frac{1}{2i \pi}\frac{d}{d\xi} \log |\xi| + \frac{1}{2} \delta(\xi)) = \frac{\log|\xi + 1/2| - \log|\xi - 1/2| }{2i \pi}+ \frac{1_{|\xi| < 1/2} }{2}$$
