Analytical transit light curves for power-law limb darkening: a comprehensive framework via fractional calculus and differential equations

We present the first complete analytical framework for computing exoplanetary transit light curves with arbitrary power-law limb darkening profiles $I(μ) \propto μ^α$, where $α$ can be any real number greater than $-1/2$, including the physically important non-integer cases. While the groundbreaking work of Agol et al. (2020) provided exact analytical solutions for polynomial limb darkening through recursion relations, stellar atmosphere models often favor power-law forms with fractional exponents (particularly $α= 1/2$) that remained analytically intractable until now. We solve this fundamental limitation through two complementary mathematical approaches: (1) Riemann-Liouville fractional calculus operators that naturally handle non-integer powers through exact integral representations, and (2) a continuous differential equation framework that generalizes discrete polynomial recursions to arbitrary real exponents. Our method provides exact analytical expressions for all half-integer powers ($α= k/2$) essential for 4-term limb darkening law by Claret (2000), maintains machine precision even at geometric contact points where numerical methods fail, and preserves the computational speed advantages crucial for parameter fitting. We demonstrate that the square-root limb darkening ($α= 1/2$) favored by recent stellar atmosphere studies can now be computed analytically with the same efficiency as traditional quadratic models, achieving 10--100$\times$ speed improvements over numerical integration while providing exact analytical derivatives.