Real Exponential Function

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 the power series $\sum _{n=0}^{\infty }\frac{x^n}{n!}$.

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

NOTATION

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

Properties

Theorem: Image of the Real Exponential Function

The image of the real exponential function is $(0;+\infty )$, i.e. its values are always positive.

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 derivative of the real exponential function is itself:

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

PROOF

TODO