On the zeros of the Scorer functions
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.
non-homogeneous Airy differential equation
properties of the zeros of the solutions
scientific article; zbMATH DE number 1891286