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Error functions, Mordell integrals and an integral analogue of a partial theta function. (English) Zbl 1393.11058

A new transformation involving the error function erf(\(z\)) and the imaginary error function erfi(\(z\)) associated with an integral analogue of a partial theta function \[ \int_0^\infty\frac{e^{-\pi\alpha^2x^2} \sin (\sqrt{\pi}\alpha xz)}{e^{2\pi x}-1}\,dx \] is established. Character analogues of this result are also given.
By repeated differentiation these transformations are used to establish results involving theta function-like integral analogues for the integral \[ \int_0^\infty\frac{xe^{-\alpha x^2}}{e^{2\pi x}-1}\,{}_1F_1(-k;3/2;2\alpha x^2)\,dx,\tag{1} \] where \({}_1F_1\) denotes the confluent hypergeometric function and \(k\) is a non-negative integer. Various exact and approximate evaluations of this integral are considered. In the case of odd \(k\) and \(\alpha=\pi\) a closed-form evaluation is given in terms of a Gauss hypergeometric function.
An asymptotic expansion for an integral involving a product of Riemann \(\Xi\)-functions of different arguments is obtained, which generalises known results of O. Oloa [Contemp. Math. 517, 305–311 (2010; Zbl 1213.33008)]. The paper concludes with some general remarks and two open questions related to the integral in (1).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
34E05 Asymptotic expansions of solutions to ordinary differential equations

Citations:

Zbl 1213.33008

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References:

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