Logarithmic Integrals

Periodizing improper integrals via the partial-fraction expansion of cosecant squared reduces trigonometric-logarithmic integrals on the half-line to finite intervals.

Poisson-kernel series expansions and two-variable dilogarithm relations evaluate a difference of arctangent integrals.

Integration by parts and the Fourier series of log tangent evaluate the inverse tangent integral at tan(pi/12) in terms of Catalan's constant.

The third part uses complex contours to evaluate logarithmic integrals involving inverse trigonometric functions.

The second part develops further logarithmic integral evaluations through polylogarithmic identities.

The first part of a series on logarithmic integrals: harmonic-number sums, polylogarithms, and residues.

Derive Kummer's Fourier series for the logarithm of the gamma function, then use it to evaluate Vardi's integral.