Certain relations for exponential-integral Bessel functions of genus 1. (English. Russian original) Zbl 0734.33002

Agrest, M. M.

Sov. Math. 35, No. 6, 67-69 (1991); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1991, No. 6(349), 67-69 (1991).

The paper deals with investigation of the so-called Bessel exponential- integral functions of the first kind \[ Ze_ 0(x,y;-1)=\sigma^ 2\int^{y}_{0}[1-e^{-xt}Z_ 0(t)]t^{-1}dt, \]
\[ Ze_ k(x,y;- 1)=\sigma^ 2\int^{y}_{0}e^{-xt}Z_ k(t)t^{-1}dt,\quad k=1,2,..., \] where \(Z_ k(t)\) is the Bessel function of the first kind \(J_ k(t)\) or its modification \(I_ k(t)\). Some relations for the above functions are proved and applications to evaluation of the integrals \[ \int^{\infty}_{0}e^{-xt} \ln tZ_ k(t)dt,\quad k=1,2,..., \] via elementary functions are indicated.

Reviewer: A.A.Kilbas (Minsk)

MSC:

33C10

Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Keywords:

Bessel function of the first kind