Exponential
The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\exp(z) = e^z = \sum_{k=0}^{\infty} \dfrac{x^k}{k!},$$ where $e$ is the base of the natural logarithm. It can be characterized as the unique solution to the initial value problem $$\left\{ \begin{array}{ll} y'=y \\ y(0)=1. \end{array} \right.$$
- Exp.png
Graph of $\exp$ on $\mathbb{R}$.
- Complex exp.jpg
Domain coloring of analytic continuation of $\exp$.
Properties
Theorem
The following formula holds: $$e^z={}_0F_0(;;z),$$ where ${}_0F_0$ denotes the hypergeometric 0F0 and $e^z$ denotes the exponential.
Proof
References
Theorem
The following formula holds for $|z|<\pi$: $$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!},$$ where $e^{xt}$ denotes the exponential function and $E_n$ denotes an Euler E polynomial.
Proof
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.14 (2)$
