Proposition: Cosine limit

On $R>0$,

Here $\Theta$ is the locally integrable Heaviside distribution; its assigned point value at zero is irrelevant.

Proof. For $R>|v|$, both arctangent arguments tend to positive infinity, so their sum tends to $\pi$. For $R<|v|$, one tends to positive infinity and the other to negative infinity, so the sum tends to zero.

For all $R>0$,

The majorant is locally integrable in $(v,\mathbf{z})\in \mathbb{R}\times {\mathbb{R}}^3$, since spatial polar measure is $R^2\mathrm{d}R\mathrm{d}\Omega$. Thus dominated convergence proves local distributional convergence, including neighbourhoods of the spatial apex. Its limit agrees with the independently defined Fourier distribution.

The radial display is used on $R>0$. At the apex the Fourier definition remains authoritative. The global convergence argument does not leave freedom to add an apex contact.