= Fourier cosine transform
{c}
{title2=$C(k)=\int_0^\infty f(x)\cos(kx)dx$}
= Cosine transform
{synonym}
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 $\widehat f_c=2C$, the inverse factor is $1/\pi$.
With boundary terms at infinity vanishing, two <integrations by parts> give the cosine transform of $f''$ as $-k^2C(k)-f'(0)$. Thus a homogeneous <Neumann boundary condition> makes it a multiplier $-k^2C(k)$, paralleling the <Fourier sine transform> with a homogeneous <Dirichlet boundary condition>.
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