A Uniform Two-Variable Analytic Approximation for the Modified Bessel Function Iν(x) in the Interval −1/2 ≤ ν ≤ 1/4

A global, uniform two-variable analytic approximation for the modified Bessel function Iν(x) is presented, valid for all real x and for orders −1/2≤ν≤1/4. The approximation is constructed using a two-variable multipoint quasi-rational approximation (MPQA) approach, in which the argument x and the order ν are treated simultaneously as independent variables. The method consistently incorporates the power-series expansion at small arguments and the asymptotic behavior at large arguments, leading to an explicit analytic representation that preserves the correct limiting behaviors. The resulting approximation remains suitable for analytical differentiation and integration, while all parameters are obtained from linear equations, avoiding numerical fitting procedures. A numerical analysis over the entire domain considered shows excellent agreement with the exact function. The largest relative error observed is εr=0.0213, occurring at ν=−0.34 and x=2.56. These results indicate that the proposed approximation provides an accurate and efficient analytic representation of Iν(x) throughout the investigated domain.