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Real Exponentiation#

The Real Exponential Function#

Theorem: The Real Exponential Function

The real power series \(\displaystyle \sum_{n = 0}^\infty \frac{x^n}{n!}\) 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.

\[ \exp(x) \overset{\text{def}}{=} \sum_{n = 0}^\infty \frac{x^n}{n!} \]

Notation

\[ \exp(x) \qquad \mathrm{e}^x \]

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.

\[ \mathrm{e}^x \lt \mathrm{e}^y \iff x \lt y \]
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

\[ (\mathrm{e}^x)' = \mathrm{e}^x \]
Proof

TODO

Theorem: Antiderivatives of the Real Exponential Function and Variations

The real exponential function \(e^{x}\) is antidifferentiable on \(\mathbb{R}\):

\[\int \mathrm{e}^x \,\mathrm{d}x = \mathrm{e}^x + C\]

The function \(\mathrm{e}^{\alpha x}\) is antidifferentiable on \(\mathbb{R}\) for all \(\alpha \ne 0\):

\[\int \mathrm{e}^{\alpha x} \,\mathrm{d}x = \frac{1}{\alpha} \mathrm{e}^{\alpha x} + C\]

If a real function \(f\) is differentiable on some open interval \(I\), then \(f'(x)\mathrm{e}^{f(x)}\) is antidifferentiable on \(I\):

\[\int f'(x)\mathrm{e}^{f(x)} \,\mathrm{d}x = \mathrm{e}^{f(x)} + C\]

If a real function \(f\) is differentiable on some open interval \(I\), then \(\mathrm{e}^x(f(x) + f'(x))\) is antidifferentiable on \(I\):

\[\int \mathrm{e}^x(f(x) + f'(x)) \,\mathrm{d}x = \mathrm{e}^x f(x) + C\]
Proof

TODO

Theorem: The Real Exponential through Convergent Sequences

The real exponential function \(e^x\) is equal to the limit of the following real sequence:

\[e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n\]
Proof

TODO

Theorem: Real Exponentiation in Bachmann-Landau

The real exponential \(\mathrm{e}^x\) can be expressed in Bachmann-Landau notation for \(x \to 0\) as follows:

\[\mathrm{e}^x = \sum_{k=0}^m \frac{x^k}{k!} + O(x^{m+1}) \qquad \text{for} \qquad x \to 0\]
Proof

TODO

Exponentiation#

Definition: Exponentiation

Let \(a\) and \(b\) be real numbers.

We define the exponentiation \(a^b\) as

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

Theorem: Antidifferentiability of Exponentiation

The exponential \(a^x\) is antidifferentiable on \(\mathbb{R}\) for all \(a \gt 0\) and for \(a \ne 1\) its antiderivatives can be constructed via the real natural logarithm:

\[\int a^x \,\mathrm{d}x = \frac{a^x}{\ln a} + C\]
Proof

TODO