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On the zeros of the Scorer functions. (English) Zbl 1010.33001

The authors develop asymptotic approximations for the zeros of the functions Gi(z) and Hi(z), where

Gi(z)=1 π 0 sin(zt+1 3t 3 )dt

and it is a particular solution of the differential equation w '' -zw=-1/π and

Hi(z)=1 π 0 e (zt-1 3t 3 ) dt

and it is a solution of w '' -zw=1/π. As it is known w '' -zw=±1/π, 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 Gi ' (z) as well as asymptotics of the negative zeros of Gi(z). They also study the complex zeros of Gi(z), Gi ' (z) and Hi(z). Tables are given with numerical values of the zeros.

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1