Six important elementary functions are the reciprocal of the exponential function:
\begin{equation*}
\exp(-z) = \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(n + 1)} z^{n}
\end{equation*}
the exponential function:
\begin{equation*}
\exp(+z) = \sum_{n = 0}^{\infty} \frac{1}{\Gamma(n + 1)} z^{n}
\end{equation*}
the trigonometric cosine and sine functions:
\begin{align*}
\cos(z) &= \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(2n + 1)} z^{2n} & \sin(z) &= \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(2n + 2)} z^{2n+1}
\end{align*}
and the hyperbolic cosine and sine functions:
\begin{align*}
\cosh(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(2n + 1)} z^{2n} & \sinh(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(2n + 2)} z^{2n+1}
\end{align*}
These functions are special cases of generalized elementary functions:
\begin{align*}
A_{pq}(z) &= \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(pn + q + 1)} z^{pn + q} & B_{pq}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(pn + q + 1)} z^{pn + q}
\end{align*}
Here is a list of blog posts associated with this project:
- Cubic Functions
- Quartic Functions 1
- Quartic Functions 2
- Hexa-Hyperbolic Functions
- Poly-Hyperbolic Sigmoids
The nomenclature is still on-going and not final.
Here is a repository with notes:
This is a work-in-progress.