From a Windowed Square Wave to the Hyperbolic Tangent, Even Zeta Values, and Dirichlet L-Values

We window a delayed periodic square wave with a decaying signum function and evaluate the product at zero frequency in both the time and frequency domains. The window renders every series that appears absolutely convergent, so the argument requires only elementary integrals and a limit. We obtain new unified derivations for the Mittag-Leffler partial-fraction expansion for the hyperbolic tangent, the Fourier expansion of the first Euler polynomial E1, and the Basel sum, zeta(2). The same calculation, followed by termwise integration, yields the Dirichlet beta value beta(3) = pi^3/32, and the values at s = 3 of two other Dirichlet L-functions. Termwise integration of this sine series produces zeta(4), and an induction continues the process to zeta(2k) for every k greater than or equal to 1.