User:Keith.U/Exposition of the Natural Exponential Function/Real

Preamble
The (real) exponential function is a real function and is denoted $\exp$.

Definition

 * $\exp: \R \to \R$ can be defined as the limit of the following power series:


 * $\exp x := \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$

Lemma: $\exp$ exists and is finite

 * $\exp x$ as defined above is well-defined.


 * Proof

Theorem: $\exp$ is continuous

 * $\exp x$ as defined above is continuous.


 * Proof

Theorem: Sum of arguments

 * $\map \exp {x + y} = \map \exp x \map \exp y$


 * Proof

Theorem: Product of arguments

 * $\map \exp {x y} = \map \exp x^y$


 * Proof

Theorem: $\exp$ is strictly positive

 * $\exp$ is strictly positive.


 * Proof

Theorem: $\exp$ is its own derivative

 * $D \exp = \exp$


 * Proof

Theorem: $\exp$ is strictly increasing

 * $\exp$ is strictly increasing


 * Proof

Definition

 * $\exp: \R \to \R$ can be defined as the limit of the following sequence:


 * $\exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$

Lemma: $\exp$ exists and is finite

 * $\exp x$ as defined above is well-defined.


 * Proof

Theorem: $\exp$ is continuous

 * $\exp x$ as defined above is continuous.


 * Proof

Theorem: Sum of arguments

 * $\map \exp {x + y} = \map \exp x \map \exp y$


 * Proof

Theorem: Product of arguments

 * $\map \exp {x y} = \map \exp x^y$


 * Proof

Theorem: $\exp$ is strictly positive

 * $\exp$ is strictly positive.


 * Proof

Theorem: $\exp$ is its own derivative

 * $D \exp = \exp$


 * Proof

Theorem: $\exp$ is strictly increasing

 * $\exp$ is strictly increasing


 * Proof

Definition

 * Let $e$ denote Euler's number.


 * $\exp: \R \to \R$ can be defined as:


 * $\exp x := e^x$
 * where $e^x$ is the unique continuous extension of the mapping $x \mapsto e^x$ from $\Q$ to $\R$.

Lemma: $\exp$ exists and is finite

 * $\exp x$ as defined above is well-defined.


 * Proof

Theorem: $\exp$ is continuous

 * $\exp x$ as defined above is continuous.


 * Proof

Theorem: Sum of arguments

 * $\map \exp {x + y} = \map \exp x \map \exp y$


 * Proof

Theorem: Product of arguments

 * $\map \exp {x y} = \map \exp x^y$


 * Proof

Theorem: $\exp$ is strictly positive

 * $\exp$ is strictly positive.


 * Proof

Theorem: $\exp$ is its own derivative

 * $D \exp = \exp$


 * Proof

Theorem: $\exp$ is strictly increasing

 * $\exp$ is strictly increasing


 * Proof

Definition

 * $\exp: \R \to \R$ can be defined as the inverse mapping of the natural logarithm $\ln$, where $\ln$ is defined as:


 * $\ds \ln x := \int_1^x \frac {\d t} t = \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$

Lemma: $\exp$ exists and is finite

 * $\exp x$ as defined above is well-defined.


 * Proof

Theorem: $\exp$ is continuous

 * $\exp x$ as defined above is continuous.


 * Proof

Theorem: Sum of arguments

 * $\map \exp {x + y} = \map \exp x \map \exp y$


 * Proof

Theorem: Product of arguments

 * $\map \exp {x y} = \map \exp x^y$


 * Proof

Theorem: $\exp$ is strictly positive

 * $\exp$ is strictly positive.


 * Proof

Theorem: $\exp$ is its own derivative

 * $D \exp = \exp$.


 * Proof

Theorem: $\exp$ is strictly increasing

 * $\exp$ is strictly increasing


 * Proof

Definition

 * $\exp: \R \to \R$ can be defined as the unique solution to the initial value problem:


 * $\dfrac {\d y} {\d x} = \map f {x, y}$
 * $\map y 0 = 1$

on $\R$, where $\map f {x, y} = y$.

Lemma: $\exp$ exists and is finite

 * $\exp x$ as defined above is well-defined.


 * Proof

Theorem: $\exp$ is continuous

 * $\exp x$ as defined above is continuous.


 * Proof

Theorem: $\exp$ is strictly positive

 * $\exp$ is strictly positive.


 * Proof

Theorem: Sum of arguments

 * $\map \exp {x + y} = \map \exp x \map \exp y$


 * Proof

Theorem: Product of arguments

 * $\map \exp {x y} = \map \exp x^y$


 * Proof

Theorem: $\exp$ is strictly increasing

 * $\exp$ is strictly increasing


 * Proof

Theorem: Equivalence of definitions of $\exp$

 * All definitions of $\exp$ hitherto are equivalent.


 * Proof