= Periodic finite-length tangent correlation of a stretched worm-like chain
{title2=$C_L(r)=\frac{k_BT}{\sqrt{Af}}\frac{\cosh[(L/2-d_L(r))/\xi]}{\sinh[L/(2\xi)]}$}
Periodizing the bulk <transverse tangent correlation of a stretched worm-like chain> gives $C_L(r)=\sum_{j\in\mathbb Z}C(r+jL)$, where $d_L(r)$ is the distance to zero on the periodic interval. Summing the geometric series yields the displayed result. At zero separation its finite-length factor is $\coth(L/(2\xi))$; for $L\gg\xi$ the bulk result is recovered. Periodic tangent fluctuations are a mathematical boundary convention, not a claim that a pulled physical chain forms a closed loop.
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