Padé Approximant of the Integral of the Scorer Hi(z) Function

This article explores a systematic and efficient method for numerically evaluating the integral of the complex Scorer Hi( z ) function, which plays a key role in solving the inhomogeneous Airy differential equation. Because this integral lacks a known closed-form expression, we introduce a Padé approximant approach to considerably boost the accuracy and speed of its evaluation. Building on existing algorithms that rely on stabilizing non-oscillating contour paths, we propose an alternative that harnesses the power of higher-order Padé approximants. We also derive a connection formula linking the integral of the Scorer Hi( z ) function to the Airy Ai( z ) function. This connection asserts a useful symmetry in the complex plane, making it easier to map out evaluation regions. In addition to this, we extend the Padé approximant insights to the Scorer function itself and its derivative, ensuring robust and reliable approximations that properly handle spurious poles. Ultimately, this work advances computational tools for dealing with special functions, making it easier to evaluate the Scorer Hi( z ) function and its integral—tools that are valuable in a range of scientific applications.