Extended Hadamard expansions for the Airy functions (Q6561295)

The paper presents a new Hadamard series expansion for the Airy function \(\operatorname{Ai}(z)\) of complex argument. This expansion is written using five series, which all include incomplete gamma functions of \(|z|\). Furthermore, taking into account that the Airy function of the second kind \(\operatorname{Bi}(z)\) can be written as\N\[\N\operatorname{Bi}(z) = e^{\pi i/6} \operatorname{Ai}(z e^{2\pi i/3}) + e^{-\pi i/6} \operatorname{Ai}(z e^{-2\pi i/3}),\N\]\Nsimilar series expansions can be obtained for \(\operatorname{Bi}(z)\).\N\NIn addition, the new expansions for \(\operatorname{Ai}(z)\) are analyzed numerically in the rectangle \(\{ z = x + yi : -10 \leq x \leq 10, -10 \leq y \leq 10\}\) of the complex plane.