Let be the series:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \sum_{n} e^{if(n)}[/tex] where f is a function perhaps a Polynomial ..then my question is..how can this series to be evaluated (at least approximately) ?..perhaps using Euler-Bernoulli sum formula, and another question what are they used for?, i heard in a book that Goldbach conjecture could be proved using them.

Another question if we have..[tex] \sum_{n} e^{if(n)}[/tex] summed over the integers or a subset of integers..could the numbers f(n) be considered "frecuencies of vibration" or eigenvalues of a certain operator?..in fact there is an interesting connection with Physics ..if we define the partition function:

[tex] Z(u)= \sum_{n>0}e^{-uE(n)} [/tex] under "complex rotation" (u-->ix ) the partition function becomes a trigonometric sum..where in this case E(n) are the "energies" (eigenvalues) of a certain Hamiltonian.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Trigonometric series

**Physics Forums | Science Articles, Homework Help, Discussion**