## ABSTRACT

Our considerations in this chapter are concerned with different types of summation formulas for the one-dimensional Euclidean space R based on the work of L.J. Mordell [1928a,b, 1929]. We start with the classical Euler summation formula for the operator of the first and second order derivative, i.e., the gradient and the Laplace operator in one dimension, respectively, and “periodical boundary conditions”. Moreover, we recapitulate the close relationship between the classical Euler summation formula and the one-dimensional Riemann Zeta function. Kronecker’s formula is mentioned, and the Theta function and its functional equation are presented. In addition, we are interested in the intimate relationship between the Euler summation formula and the Poisson summation formula interconnected by properties of the Laplacian.