Analytical Transit Light Curves for Arbitrary Power-Law Limb Darkening: A Unified Framework

We present the first exact analytical solutions for exoplanet transit light curves under arbitrary real power-law limb darkening, $I(\mu) = I_0\mu^\alpha$ with $\alpha>-2$, removing the two-decade restriction to integer-polynomial forms. This generalization addresses a fundamental barrier: physically motivated stellar atmosphere models require non-integer exponents -- most critically the square-root law ($\alpha=1/2$) essential for M-dwarf characterization and half-integer powers in Claret's four-parameter law -- yet existing frameworks support only integer powers, forcing reliance on numerical integration or polynomial approximations that introduce systematic errors at the $\sim 10^{-3}$ level. We develop three mathematically equivalent formulations: (1) a geometric kernel representation expressing blocked flux as a one-dimensional radial integral; (2) a fractional-calculus framework yielding exact hypergeometric solutions via Riemann-Liouville operators; and (3) a numerically stable hypergeometric-integrand form with guaranteed convergence. All three reproduce established elliptic-integral expressions for integer $\alpha$, confirming our framework as a true generalization. For central transits, we obtain the exact closed form $\mathcal{F} = (1-p^2)^{1+\alpha/2}$. Linear superposition enables exact treatment of multi-parameter laws without special-case logic. Validation using 45-digit precision arithmetic demonstrates inter-method agreement at $|\Delta\mathcal{F}| \lesssim 10^{-13}$ across all geometric regions, including transcendental exponents ($\pi/2$, $\sqrt{2}$, $e-2$). Compared with Monte Carlo integration, our method achieves nine orders of magnitude improvement in accuracy with $\sim 10^4\times$ computational speedup, enabling efficient Bayesian inference for JWST, TESS, and next-generation facilities.