Planar freely jointed chain ($L/(Nb)=I_1(\xi )/I_0(\xi )$)

If each link is restricted to a plane, $Z_1={\int }_0^{2\pi }{e}^{\xi \cos \theta }d\theta =2\pi I_0(\xi )$, with $\xi =Fb/(k_{B}T)$. Differentiating gives $L/(Nb)=I_1(\xi )/I_0(\xi )$, where $I_n$ are Modified Bessel functions of the first kind. The high-force angular Gaussian has $⟨{\theta }^2⟩\sim 1/\xi$, so $Nb-L\sim N{k}_{B}T/(2F)$, half the three-dimensional coefficient.