= Magnetic pressure support with vanishing axial field
With no axial <electric current>, <cylindrical magnetostatic pressure balance> reduces to constant $p+B_z^2/(8\pi)$. For $p=p_0e^{-R^2/a^2}$ and $B_z(0)=0$, its radial solutions are
$$
B_z=\sigma\sqrt{8\pi p_0}\sqrt{1-e^{-R^2/a^2}},
\qquad
j_\phi=-\sigma\frac{c\sqrt{8\pi p_0}}{4\pi a^2}
\frac{R e^{-R^2/a^2}}{\sqrt{1-e^{-R^2/a^2}}},
\qquad \sigma\in\{-1,1\}.
$$
As $R\downarrow0$, $B_z\sim\sigma\sqrt{8\pi p_0}R/a$ and $j_\phi\to-\sigma c\sqrt{8\pi p_0}/(4\pi a)$. These finite scalar limits do not give a <smooth function> of Cartesian position: $R=\sqrt{x^2+y^2}$ has a cusp at the axis, and the azimuthal unit vector has no unique limit there. Thus these data describe radial equilibrium away from the axis, but cannot satisfy smooth vector-field regularity on the axis without modifying the data.
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