Sums of Exponential Terms, Conserved Quantities, and the Real Wave Numbers
Terence R. Smith
TL;DR
This work addresses the challenge of representing sums of complex exponential terms $S_n = \sum_{k=1}^n R_{0k} e^{i \theta_k}$ by developing recursive, closed-form representations for any finite $n$ in the canonical form $S_n = A_n e^{i \sum_{k=1}^n \theta_k}$. It derives recursive relations for $A_n$, such as $A_0=0$ and $A_n = A_{n-1} e^{-i \theta_n} + R_{0n} e^{-i \sum_{j=1}^{n-1} \theta_j}$, and provides alternative decompositions that involve cosine nesting and a two-exponential factorization. The paper then generalizes the complex-number framework to a real wave-number field $\mathbb{W}$ generated by $\mathbf w(f, \theta)$, forming a field under pointwise operations with a multiplicative group structure $\mathbf w(f_1,\theta_1) \otimes \mathbf w(f_2,\theta_2) = \mathbf w(f_1+f_2, \theta_1+\theta_2)$ and a corresponding additive group. This construction leads to interpreting the exponential term $e^{i \sum \theta_k}$ as a projection of conserved quantities from a richer space of numbers onto the complex plane, linking sums of exponentials to invariants in an extended algebraic framework and suggesting broader implications for understanding exponential-sum representations. Overall, the results connect closed-form representations of exponential sums to a generalized algebra of real wave numbers, offering a structural perspective on invariants and projections in such sums.
Abstract
There is consensus that sums $S_n={ {Σ}_{k=1}^n R_{0k} e^{i θ_k}}$ 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 $S_n= A_n exp({ iΣ_{k=1}^n θ_k})$. The goal of the paper is to show that one may interpret the exponential term $exp(i Σ_{k=1}^n θ_k)$ 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.