Chapter 7.12, Problem 20P

In Problems 17 to 20, find the Fourier sine transform of the function in the indicated problem, and write $f(x)$ as a Fourier integral [use equation (12.14)]. Verify that the sine integral for $f(x)$ is the same as the exponential integral found previously.

Problem 12.

In Problems 3 to 12, find the exponential Fourier transform of the given $f(x)$ and write $f(x)$ as a Fourier integral [that is, find $g(\alpha )$ in equation (12.2) and substitute your result into the first integral in equation $(12.2)]$

$f(x)=$ { $\sin x,\left|x\right|<\pi /2$ $0,\left|x\right|>\pi /2$