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Here $q=\kappa /R$, so the equation from part (b) becomes the Euler-Cauchy equationThe power-law ansatz $B_z=R^s$ gives the indicial equationFor $0<\kappa <1/2$, the general field is thereforeThe polynomial discriminant changes sign atFor $\kappa >{\kappa }_c$, define $\mu =\sqrt{{\kappa }^2-1/4}$. A real form of the solution isThus the field has a log-periodic oscillation: its phase is periodic in $\ln R$, so it oscillates as the radius changes geometrically.

At $\kappa ={\kappa }_c=1/2$, the indicial equation has the multiple root $s=-1/2$. The general axial component isSince $B_{\varphi }=-2R{B}_z^{\prime }$, the azimuthal component isConsequentlyas $R\to \infty$; the same ratio is identically one when $D=0$. A magnetic field line has tangent parallel to $\mathbf{B}$, so its asymptotic angle $\vartheta$ with the $z$-axis satisfiesHence the field lines become helices making the constant anglewith the axis.