Question: Show that the Fourier sine transform pair as given in mathematical handbooks (for example, Spiegel, 1973), [ f(x)=x^{a-1} quad(0

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 $a$ :

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

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