Tagged: fourier-series

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.

Residue calculus on a rectangular contour shows that the hyperbolic secant is its own Fourier transform up to scaling.

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

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

From the defining recursion for Bernoulli polynomials to their Fourier expansions and an associated integral.

Weyl's criterion connects uniform distribution modulo one with exponential sums and averages of Riemann-integrable functions.