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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2510.07803 |
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| _version_ | 1866911199801638912 |
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| author | Smith, Terence R. |
| author_facet | Smith, Terence R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums of Exponential Terms, Conserved Quantities, and the Real Wave Numbers Smith, Terence R. Number Theory 11A67 (Primary) 30B50, 42A24 (Secondary) 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. |
| title | Sums of Exponential Terms, Conserved Quantities, and the Real Wave Numbers |
| topic | Number Theory 11A67 (Primary) 30B50, 42A24 (Secondary) |
| url | https://arxiv.org/abs/2510.07803 |