Subsection 6.1.4 The (Infinite) Fourier Sine Transform of \(\mathcal{F}(k)\)
If
\begin{equation*}
F_{s}(k) = \mathscr{F}_{s}\{f(x)\} = \int\limits_{0}^{\infty}f(x)\sin kx \, dx
\end{equation*}
then \(F_{s}(k)\) is called the Fourier sine transform of \(f(x)\) and if
\begin{equation*}
f(x) = \mathscr{F}^{-1}_{s}\{F_{s}(k)\} =\frac{2}{\pi}\int\limits_{0}^{\infty}F_{s}(k)\sin kx \, dk
\end{equation*}
is called the inverse Fourier sine transform of \(F_{s}(k)\text{.}\) Also, Fourier Cosine Transform and Inverse Fourier Cosine Transform are given as
\begin{equation*}
F_{c}(k) = \mathscr{F}_{c}\{f(x)\} = \int\limits_{0}^{\infty}f(x)\cos kx \, dx
\end{equation*}
and
\begin{equation*}
f(x) = \mathscr{F}^{-1}_{c}\{F_{c}(k)\} =\frac{2}{\pi}\int\limits_{0}^{\infty}F_{c}(k)\cos kx \, dk.
\end{equation*}
