Fourier sine transform
= Fourier sine transform
{c}
{title2=$S(k)=\int_0^\infty\sin(kx)f(x)dx$}
= Sine transform
{synonym}
The Fourier sine transform is a half-line transform adapted to a homogeneous <Dirichlet boundary condition>. With the displayed normalization, inversion is $f(x)=(2/\pi)\int_0^\infty\sin(kx)S(k)dk$. It is equivalent to the <Fourier transform> of an odd extension. If $f(0)=0$ and boundary terms vanish, two <integrations by parts> give the transform of $f''$ as $-k^2S(k)$.