A Closed-Form Solution to the Arbitrary Order Cauchy Problem with Propagators

The general abstract arbitrary order (N) Cauchy problem was solved in a closed form as a sum of exponential propagator functions. The infinite sparse exponential series was solved with the aid of a homogeneous differential equation. It generated a linear combination of exponential functions. The Cauchy problem solution was formed with N linear combinations of N exponential propagators.Comment: 6 page

On Solving Some Trigonometric Series

This communication shows the track for finding a solution for a sin(kx)/k**2 series and a fresh representation for the Euler's Gamma function in terms of Riemann's Zeta function. We have found a new series expression for the logarithm as a side effect. The new series are useful both for analysis, approximations and asymptotic studies.Comment: 12 pages, 2 figure