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Zbl 1010.33001
Gil, Amparo; Segura, Javier; Temme, Nico M.
On the zeros of the Scorer functions. (English)
[J] J. Approximation Theory 120, No.2, 253-266 (2003). ISSN 0021-9045

The authors develop asymptotic approximations for the zeros of the functions Gi$(z)$ and Hi$(z)$, where $$ \text{ Gi}(z)=\frac{1}{\pi}\int_{0}^{\infty} \sin(zt+\frac{1}{3}t^{3}) dt $$ and it is a particular solution of the differential equation $w^{\prime \prime}-zw=-1/\pi$ and $$ \text{Hi} (z)=\frac{1}{\pi}\int_{0}^{\infty} e^{(zt-\frac{1}{3}t^{3})} dt $$ and it is a solution of $w^{\prime \prime}-zw=1/\pi$. As it is known $w^{\prime \prime}-zw= \pm 1/\pi$, is the non-homogeneous Airy differential equation and the functions Gi$(z)$ and Hi$(z)$ are called the Scorer functions. The authors study qualitative properties of the real zeros of Gi$(z)$ and $\text{ Gi}'(z)$ as well as asymptotics of the negative zeros of Gi$(z)$. They also study the complex zeros of Gi$(z)$, $\text{Gi}'(z)$ and Hi$(z)$. Tables are given with numerical values of the zeros.
[Stamatis Koumandos (Nicosia)]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: non-homogeneous Airy differential equation; properties of the zeros of the solutions; Scorer functions

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