Integrals, series,
and the ideas between.

Reflecting the interval of integration and applying Euler's reflection formula evaluates a definite integral involving a fifth root.

A homographic change of variables reduces a beta-type integral with a shifted linear denominator to Euler's beta function.

A change of variables and an inversion symmetry reduce integral 3.255 from Gradshteyn and Ryzhik to a beta integral.

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

Continue the study of theta functions with infinite products and connections to elliptic integrals.

Begin with Jacobi's theta series and derive their periodicity, zeros, and fundamental identities.

A proof of the classification of integers whose multiplicative group of units is cyclic.

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.