Tagged: 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.

Differentiating the Mellin transform of the sine function with respect to its exponent evaluates a logarithmic Fresnel-type integral in terms of Euler's constant.

Integrating a complex power around a keyhole cut along the negative real axis evaluates a rational integral involving the sine of a logarithm.

A contour argument gives integral representations involving the reciprocal beta function and their logarithmic consequences.

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.

Generating functions for squared harmonic numbers provide evaluations of Euler sums and logarithmic integrals.