By H. M. Srivastava
Zeta and q-Zeta capabilities and linked sequence and Integrals is a completely revised, enlarged and up-to-date model of sequence linked to the Zeta and comparable services. some of the chapters and sections of the ebook were considerably changed or rewritten and a brand new bankruptcy at the thought and functions of the elemental (or q-) extensions of varied exact services is integrated. This booklet can be useful because it covers not just precise and systematic shows of the speculation and purposes of many of the tools and strategies utilized in facing many alternative sessions of sequence and integrals linked to the Zeta and comparable services yet stimulating ancient money owed of a big variety of difficulties and well-classified tables of sequence and integrals.Detailed and systematic shows of the speculation and purposes of a few of the tools and methods utilized in facing many various periods of sequence and integrals linked to the Zeta and similar features
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Additional info for Zeta and q-Zeta Functions and Associated Series and Integrals (Elsevier Insights)
5 1 in (20), and using (6) and (8), we obtain 2 1 3 G 4 4 1 1 3 9 = 2− 4 ·π − 2 · e 8 · A− 2 . 848718 · · · . (24) It follows from (23), (24) and (6) that G G 3 4 5 4 1 1 G = 2− 8 · π − 4 · e 2π . (25) Introduction and Preliminaries 45 Integral Formulas Involving the Double Gamma Function We begin by recalling the following integral formula: z πt cot π t dt = z log(2π) + log 0 G(1 − z) , G(1 + z) (26) which is due, originally, to Kinkelin . Indeed, in view of (3), if we set (z) := G(1 + z) = (2π)z e−z G(1 − z) ∞ k=1 1 + kz 1 − kz k e−2z and differentiate logarithmically with respect to z, and apply the known expansion (see Ahlfors [13, p.
Some simple consequences of Kinkelin’s formula (26) are worthy of note here (see also Barnes [94, p. 279]). First, by using integration by parts in (26), we have z log sin πt dt = z log 0 which, upon setting t = z 1 2 sin πz G(1 + z) + log , 2π G(1 − z) − u and replacing z by 1 2 (28) − z, yields cos π z 1 1 log − log 2 − log log cos π t dt = z − 2 2π 2 0 G 1 − z + log 2 G 1 2 +z 1 2 −z . (29) Making use of (29), we obtain the following analogue of (26): z 0 1 cos πz π t tan πt dt = − log − z log(2π) − log 2 π G 1 − z + log 2 G 1 2 +z 1 2 −z , (30) 46 Zeta and q-Zeta Functions and Associated Series and Integrals which would follow also from (26) by setting t = 12 − u and replacing z by Combining (28) and (29), we readily have the integral formula: z log tan πt dt = z log tan πz + 0 1 cos π z log + log 2 π G G (1 + z) + log − log G (1 − z) G 1 2 +z 1 2 −z 1 2 − z.
269]) that G (z + 1)/G(z + 1) is the only solution of f (z + 1) = f (z) + (z) , (z) which has such an asymptotic expansion near at infinity. 3(58), reduces at once to (n + 1) log n − n + z (log n + 1 + γ − 2β) + 1 − α + O n−1 2 (n → ∞), which, upon putting α= 1 1 1+γ − log (2π) and β = , 2 2 2 becomes the right-hand side of (7). (12) Introduction and Preliminaries 43 If we now substitute the values of α and β from (12) into (10), we obtain the following expression: − G (z + 1) 1 1 = − log(2π) + (1 + γ ) z G(z + 1) 2 2 + d log dz ∞ k=0 1 (z + 1 + k) exp −z ψ(1 + k) − z2 ψ (1 + k) (1 + k) 2 (|z| < 1; z ∈ C), which, upon integrating with respect to z and obtaining the value of the constant of integration by making z = 0, finally yields the desired expression (4) valid for |z| < 1.