Periodic finite-length tangent correlation of a stretched worm-like chain ($C_L(r)=\frac{k_{B}T}{\sqrt{Af}}\frac{\cosh [(L/2-d_L(r))/\xi ]}{\sinh [L/(2\xi )]}$)

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.