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Outside a core of radius $r_0$, set $\lambda \simeq {\lambda }_0$ to leading order. The uniform-state bulk energy cancels on subtraction, while the divergence elastic density is $K{\lambda }_0^2/(2r^2)$. Its two-dimensional area integral isThus the logarithmic energy of a planar nematic disclination givesThe core suppresses the nematic amplitude to regularize the singular director texture. Balancing lost bulk ordering energy against elasticity gives $r_0$ of order $\sqrt{K/|a|}$. The exact core radius and additive core energy depend on the amplitude profile.

The “local” amplitude disturbance need not be literally compactly supported. The radial Euler-Lagrange equation following from the PDF density isIts far-field expansion is $\lambda ={\lambda }_0[1-K/(2|a|r^2)+O(r^{-4})]$. These algebraic tails change only finite contributions, not the logarithm. A regular core has $\lambda =O(r^2)$ for this integer winding.

The large cutoff $L$ is set by the sample size for an isolated defect with corresponding imposed far-field winding, or by the distance to compensating defects or other texture-screening boundaries. An isolated nonzero winding in an infinite ordered plane has a logarithmically divergent excess energy; in a neutral pair, far-field distortions cancel beyond the separation. If $L$ is not much larger than $r_0$, the leading logarithm is not a dominant approximation.