The discussion revolves around alternative representations of the function frac{x}{e^{x}-1}, specifically seeking a trigonometric form without using Bernoulli numbers. Participants explore various mathematical approaches and concepts related to this function.
frac{x}{e^{x}-1} as a trigonometric function without involving Bernoulli numbers.e^x = cos(x) + i sin(x), but later states this approach is not valid.x with ix as a potential method.e^{x}-1 and its relation to frac{e^{x}-1}{x}, presenting it as a series.f(x)=x and a local linearization of e^x -1, suggesting the use of Taylor series for arcsin or arccos.frac{x}{e^{x}-1} and references Faulhaber formulas.B#1, and provides a link to the Riemann zeta function for further context.Participants present multiple competing views and approaches without reaching a consensus on a specific trigonometric representation or the role of Bernoulli numbers.
Some participants express uncertainty regarding the validity of certain mathematical identities and the implications of using trigonometric forms. The discussion includes references to series expansions and specific mathematical concepts that may require further exploration.
Regardless of what it means, writing it this way would not help since it is not true. It is ##e^x##, not ##e^{ix}## ...HallsofIvy said:I'm not sure what you mean by "display as a trigonometric function". Would writing [itex]e^x[/itex] as [itex]cos(x)+ i sin(x)[/itex] help?
This may possibly help: https://en.wikipedia.org/wiki/Riemann_zeta_function.eaglechief said:Second, i wonder why they "appear" only with 2n index (except B#1).