Natural Exponential Function

analysis

Definition

Natural Exponential Function

Die natural exponential function is defined as:

$$\exp (x)=e^x=\underset{n\to \infty }{\lim }{\left(1+\frac{x}{n}\right)}^n$$

with $e$ being Euler’s number.

Power Series Representation

The natural exponential function can be represented as power series:

$$\exp (x)=e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots =\sum _{n\ge 0}\frac{x^n}{n!}$$

Properties

Identity

$$\forall x\in \mathbb{R}:\exp (x)\cdot \exp (-x)=1$$

Homomorphic

$$\forall x\in \mathbb{R}:\exp (x+y)=\exp (x)\cdot \exp (y)$$

Thus, the natural exponential function $\exp$ is a homomorphism from the additive group $(\mathbb{R},+)$ to the multiplicative group $({\mathbb{R}}_{>0},\cdot )$:

$$\exp :(\mathbb{R},+)\to ({\mathbb{R}}_{>0},\cdot )$$

Limit

$$\underset{n\to \infty }{\lim }{a}_n=a⟹\underset{n\to \infty }{\lim }E(a_n)=E(a)$$