Exact analytical solutions of the inverse Langevin function with application to non-Newtonian rheology

The inverse Langevin function (ILF) is central to finite-extensibility models in polymer rheology, magnetism, and nonlinear elasticity, yet its singular behavior and lack of a closed form have kept approximations in daily use that trade accuracy for simplicity. We contribute two advancements to the field: (1) an exact analytical solution over the full domain and (2) an analytical solution that is exact within the radius of convergence of the Taylor series solution, accurate to double precision (and beyond) across the full physical domain, and asymptotically consistent at both ends of the domain. The former provides a full analytic continuation of the power series representation of the ILF and is of theoretical importance, having slow convergence properties. The latter is an asymptotically consistent resummation (ACR) that is comparable in accuracy and efficiency to the spline of Benítez & Montáns (2018). We evaluate the proposed resummation across both continuum closure-based (Peterlin) and closure-free (stochastic Brownian dynamics) modeling frameworks for bead–spring dumbbell ensembles. In both simple shear and extensional flows, the new form eliminates mathematical truncation errors, matching a numerically exact ILF reference solution to within numerical tolerance and significantly outperforming rational forms (e.g., Warner, Cohen) under strong stretching regimes. Crucially, standalone CPU benchmarks demonstrate that this mathematical fidelity is achieved with only modest computational overhead relative to simple rational approximations while being substantially faster than iterative solvers, making our solution a practical drop-in replacement for advanced microstructural simulations.