The inverse tangent integral and its association with Euler sums

(zeta (s)) and (L(s, chi)) (11M06) Gamma, beta and polygamma functions (33B15) Integration of real functions of several variables: length, area, volume (26B15) Hurwitz and Lerch zeta functions (11M35) Numerical summation of series (65B10) Trigonometric moment problems in one variable harmonic analysis (42A70)

Cites Work

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Explicit evaluation of Euler and related sums
Integer powers of arcsin
On some series containing \(\psi{}(x)-\psi{}(y)\) and \((\psi{}(x)- \psi{}(y))^ 2\) for certain values of \(x\) and \(y\)
(Almost) impossible integrals, sums, and series. With a foreword by Paul J. Nahin
General order Euler sums with multiple argument
Inside interesting integrals (with an introduction to contour integration). A collection of sneaky tricks, sly substitutions, and numerous other stupendously clever, awesomely wicked, and devilishly seductive maneuvers for computing hundreds of perplexing definite integrals from physics, engineering, and mathematics (plus numerous challenge problems with complete, detailed solutions)
Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers
Extension of the four Euler sums being linear with parameters and series involving the zeta functions
Log-hyperbolic tangent integrals and Euler sums
Four parametric linear Euler sums
Log-sine-polylog integrals and alternating Euler sums
A family of shifted harmonic sums
Central Binomial Sums, Multiple Clausen Values, and Zeta Values
Extensions of Euler harmonic sums
Log-Sine integrals involving series associated with the zeta function and polylogarithms
EVALUATING LOG-TANGENT INTEGRALS VIA EULER SUMS
Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log-sine integrals
A Definite Integral of a Product of Two Polylogarithms
Interesting Series Involving the Central Binomial Coefficient
Euler Sums and Contour Integral Representations
A restricted Epstein zeta function and the evaluation of some definite integrals
Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums
Explicit evaluation of Euler sums
On the Evaluation of Euler Sums
Euler-like sums via powers of log, arctan and arctanh functions
The Riemann zeta function and classes of infinite series
Families of specialized Euler sums
Explicit evaluation of some integrals involving polylogarithm functions
A natural companion to Catalan's constant
Alternating Euler sums and BBP-type series
Parameterized families of polylog integrals
General order Euler sums with rational argument
Some evaluations of infinite series involving parametric harmonic numbers
Integrals of logarithmic functions and alternating multiple zeta values
Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms
Closed Forms: What They Are and Why We Care
SHIFTED HARMONIC SUMS OF ORDER TWO
EULER SUMS EVALUATABLE FROM INTEGRALS
Special Integrals of Gradshteyn and Ryzhik
Special Integrals of Gradshteyn and Ryzhik

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