## A type of generalized trigonometric functions and some non-interesting applications

#### Han Yu

Trigonometric functions  e.g. $$sin$$, $$cos$$...are very interesting, simple and full of magic. If we consider them as a pair of functions derived from the exponential function in a certain way, it is then not very un-natural to generalized the idea further, this will give us triple,quadruple, 5-ple...functions, those functions are less interesting, less simple, and not that full of magic but still, there will be applications such as evaluate series $$\sum_{i=0}^{\infty} \frac{n^k}{n^{2m}+1}$$ and $$\sum_{i=0}^{\infty} \frac{n^k}{n^{2m}+1}$$ for quite a lot of numbers $$k$$ and $$m$$. In the seminar, the construction of generalized trigonometric functions and the evaluation of the series will be talked about.