= Free-end curvature boundary layer
{title2=$\ell=\sqrt{B/F}$}
Moment-free ends require $\zeta''=\kappa_0$ there, so a highly stretched finite filament can retain curvature in boundary layers of width $\ell=\sqrt{B/F}$. For constant intrinsic curvature $\kappa$, zero transverse end forces give slope $\zeta'(x)=\kappa\ell\sinh[(x-L/2)/\ell]/\cosh[L/(2\ell)]$. Direct integration yields
$$
\mathcal L-L=\frac{\kappa^2\ell^3}2\left[\tanh a-a\operatorname{sech}^2a\right],\qquad a=L/(2\ell).
$$
Its large-force deficit is $\kappa^2\ell^3/2$, of order $F^{-3/2}$, even though the smooth bulk curvature derivative vanishes. This shows why endpoint assumptions matter in <force-extension of an intrinsically curved filament>.
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