Lemma: Global radial representatives

The cosine distribution is the locally integrable function

Moreover $D=-{\partial }_{v}H$. For $R>0$,

The derivative of (eq:H) specifies the distribution at the spatial apex, with no added point-supported term.

Proof. Angular integration gives

For real $a$,

as follows by differentiating in $a$, integrating the elementary Laplace transform, and using the value zero at $a=0$. Consequently

It converges almost everywhere to (eq:H). For $R>0$, its absolute value is at most $1/(4\pi R)$, which is locally integrable in $(v,\mathbf{z})$. Dominated convergence gives the global distributional limit, including $R=0$.

Fourier differentiation gives $D=-{\partial }_{v}H$. Differentiating the indicator at $R>0$ gives (eq:D). Since the global limit of $H_{ϵ}$ has already been identified as a locally integrable function, its distributional derivative leaves no undetermined apex contact.