Show that the Fourier sine transform pair as given in mathematical handbooks (for example, Spiegel, 1973),

\[
f(x)=x^{a-1} \quad(0\]

reduces to the following transform pair for an appropriate choice of \(a\) :

\[
f(x)=\frac{1}{\sqrt{x}} \quad \text { and } \quad F(w) \equiv \mathscr{F}\{f(x)\}=\frac{1}{\sqrt{w}}
\]