Sums of Exponential Terms, Conserved Quantities, and the Real Wave Numbers

There is consensus that sums $S_n={ {\Sigma }_{k=1}^n R_{0k} e^{i \theta_k}}$ of complex exponential terms, despite their mathematical significance, only possess closed-form representations for specific values of n and special values of their parameters and that there are no generally-accepted recursive formulae for their computation. This note is focused on recursive formulae that: (1) provide closed-form analytic representations of $S_n$ for any finite n; (2) include generalizations of the usual formula for the sum of two exponentials; and (3) are representable in the form $S_n= A_n exp({ i\Sigma_{k=1}^n \theta_k})$. The goal of the paper is to show that one may interpret the exponential term $exp(i \Sigma_{k=1}^n \theta_k)$ of $S_n$ as representing the projection, from a field of numbers that generalizes the complex numbers onto the complex plane, of a term representing quantities that are conserved under the addition and multiplication of numbers in the extended space. In particular, it is shown that the general form of a number in the extended field generalizes the form of a sum of complex exponentials.