Real Exponentiation

The Real Exponential Function

Theorem: The Real Exponential Function

The real power series $\sum _{n=0}^{\infty }\frac{x^n}{n!}$ converges for all $x\in \mathbb{R}$.

PROOF

TODO

Definition: The Real Exponential Function

The real exponential function is the real analytic function $\exp :\mathbb{R}\to \mathbb{R}$ defined by this real power series.

$$\exp (x)\stackrel{\text{def}}{=}\sum _{n=0}^{\infty }\frac{x^n}{n!}$$

NOTATION

$$\exp (x){\mathrm{e}}^x$$

Theorem: Image of the Real Exponential Function

The image of the real exponential function is the open interval $(0;+\infty )$.

PROOF

TODO

Theorem: Monotony of the Real Exponential Function

The real exponential function is strictly increasing.

$${\mathrm{e}}^x<{\mathrm{e}}^y⟺x<y$$

PROOF

TODO

Addition Theorem for the Real Exponential Function

The real exponential function has the following property for all $x,y\in \mathbb{R}$:

$${\mathrm{e}}^x{\mathrm{e}}^y={\mathrm{e}}^{x+y}$$

PROOF

TODO

Theorem: Derivative of the Real Exponential Function

The real exponential function is differentiable with

$$({\mathrm{e}}^x{)}^{\prime }={\mathrm{e}}^x$$

PROOF

TODO

Theorem: Antiderivatives of the Real Exponential Function

The antiderivatives of the real exponential function are given by

$$\int {\mathrm{e}}^x\mathrm{dx}={\mathrm{e}}^x+\text{const}$$

PROOF

TODO

Exponentiation

Definition: Exponentiation

Let $a$ and $b$ be real numbers.

We define the exponentiation $a^b$ as

a^b \overset{\text{def}}{=} \left\{ \begin{array}{l@{\quad}l} 1 & \text{if } a \ne 0, b = 0 \\ \underset{b\text{ times}}{\underbrace{a \times \cdots \times a}} & \text{if } b \in \mathbb{N} \\ \sqrt[n]{a^m} & \text{if } b = \frac{m}{n} \text{ with } m,n \in \mathbb{N} \end{array} \right.