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Integral representations of quantities associated with Gamma functionстатья
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Дата последнего поиска статьи во внешних источниках: 20 апреля 2022 г.
- Авторы: Kostin A.B., Sherstyukov V.B.
- Журнал: Ufa Mathematical Journal
- Том: 13
- Номер: 4
- Год издания: 2021
- Первая страница: 50
- Последняя страница: 62
- DOI: 10.13108/2021-13-4-50
- Аннотация: We study a series of issues related with integral representations of Gammafunctions and its quotients. The base of our study is two classical results in the theoryof functions. One of them is a well-known first Binet formula, the other is a less knownMalmsten formula. These special formulae express the values of the Gamma function inan open right half-plane via corresponding improper integrals. In this work we show thatboth results can be extended to the imaginary axis except for the point 𝑧 = 0. Undersuch extension we apply various methods of real and complex analysis. In particular, weobtain integral representations for the argument of the complex quantity being the value ofthe Gamma function in a pure imaginary point. On the base of the mentioned Malmstenformula at the points 𝑧 ̸= 0 in the closed right half-plane, we provide a detailed derivationof the integral representation for a special quotient expressed via the Gamma function:𝐷(𝑧)≡Γ(𝑧+1/2)/Γ(𝑧+1). This fact on the positive semi-axis was mentioned without theproof in a small note by Duˇsan Slavi´c in 1975. In the same work he provided two-sidedestimates for the quantity 𝐷(𝑥) as 𝑥 > 0 and at the natural points 𝐷(𝑥) coincided withthe normalized central binomial coefficient. These estimates mean that 𝐷(𝑥) is envelopedon the positive semi-axis by its asymptotic series.In the present paper we briefly discuss the issue on the presence of this property on theasymptotic series 𝐷(𝑧) in a closed angle |arg 𝑧|<= 𝜋/4 with a punctured vertex. By thenew formula representing 𝐷(𝑧) on the imaginary axis we obtain explicit expressions for thequantity |𝐷(𝑖𝑦)| and for the set Arg 𝐷(𝑖𝑦) as 𝑦 > 0. We indicate a way of proving thesecond Binet formula employing the technique of simple fractions.
- Добавил в систему: Шерстюков Владимир Борисович