Exponential Taylor Series

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda \in \mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an infinite series to $C^{\mathrm{\infty }}$ functions, which is shown to recover the original function under suitable conditions on the remainder. These results are also used to calculate some infinite series involving Stirling Numbers, as well as providing a few examples.