 Fourier cosine transform

For sufficiently regular decaying data, this half-line transform has inverse $f(x)=(2/\pi ){\int }_0^{\infty }C(k)\cos (kx)dk$. To derive it, extend $f$ evenly to the real line. Its Fourier transform with negative exponential is $2C(k)$, and its odd sine contribution vanishes. The Fourier inversion theorem then gives the stated factor $2/\pi$. If the forward transform is instead defined as ${\hat{f}}_c=2C$, the inverse factor is $1/\pi$.

With boundary terms at infinity vanishing, two integrations by parts give the cosine transform of $f^{\prime \prime }$ as $-k^{2}C(k)-f^{\prime }(0)$. Thus a homogeneous Neumann boundary condition makes it a multiplier $-k^{2}C(k)$, paralleling the Fourier sine transform with a homogeneous Dirichlet boundary condition.