Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results

DOI10.1007/s11139-013-9528-5zbMath1336.33041OpenAlexW1980307955WikidataQ29012275 ScholiaQ29012275MaRDI QIDQ2254851

Iaroslav V. Blagouchine

Publication date: 6 February 2015

Published in: The Ramanujan Journal (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11139-013-9528-5

zbMATH Keywords

gamma functionlogarithmic integralorthogonal expansionszeta functiongeneralized Euler's constantslogarithmic seriesspecial constants

Mathematics Subject Classification ID

(zeta (s)) and (L(s, chi)) (11M06) Gamma, beta and polygamma functions (33B15) Other functions defined by series and integrals (33E20) History of mathematics in the 19th century (01A55) Hurwitz and Lerch zeta functions (11M35) Representations of entire functions of one complex variable by series and integrals (30D10)

Related Items (12)

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to \(\pi^{-1}\) ⋮ A series representation of Euler-Stieltjes constants and an identity of Ramanujan ⋮ A master integral in four parameters ⋮ Unnamed Item ⋮ Integral representations of quantities associated with Gamma function ⋮ Paperfolding infinite products and the gamma function ⋮ A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations ⋮ Three notes on Ser's and Hasse's representations for the zeta-functions ⋮ Some new sums of q-trigonometric and related functions through a theta product of Jacobi ⋮ Shifted Euler constants and a generalization of Euler-Stieltjes constants ⋮ Euler and the Cubic Basel Problem ⋮ Another Alternating Analogue of Euler’s Constant

Uses Software

• OEIS

Cites Work

• Series representations for the Stieltjes constants
• On representations and differences of Stieltjes coefficients, and other relations
• Some constants associated with the Riemann zeta-function
• A class of logarithmic integrals
• Derivatives of the Hurwitz zeta function for rational arguments
• Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz
• The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa
• The integrals in Gradshteyn and Rhyzik. Part 1: A family of logarithmic integrals
• The Gamma Function and the Hurwitz Zeta-Function
• Integrals, an Introduction to Analytic Number Theory
• AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
• On Some Generalizations of the Cauchy-Frullani Integral
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