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:


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

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

 * $\exp \left({ x + y }\right) = \exp \left({ x }\right) \exp \left({ y }\right)$


 * Proof

Theorem: Product of arguments

 * $\exp \left({ xy }\right) = \exp \left({ x }\right)^{y}$


 * 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:


 * $\exp x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$

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

 * $\exp \left({ x + y }\right) = \exp \left({ x }\right) \exp \left({ y }\right)$


 * Proof

Theorem: Product of arguments

 * $\exp \left({ xy }\right) = \exp \left({ x }\right)^{y}$


 * 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

 * $\exp \left({ x + y }\right) = \exp \left({ x }\right) \exp \left({ y }\right)$


 * Proof

Theorem: Product of arguments

 * $\exp \left({ xy }\right) = \exp \left({ x }\right)^{y}$


 * 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:


 * $\displaystyle \ln x := \int_1^x \frac {\mathrm dt} t = \lim_{n \to \infty} n \left({ \sqrt[n]{ x } - 1 }\right)$

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

 * $\exp \left({ x + y }\right) = \exp \left({ x }\right) \exp \left({ y }\right)$


 * Proof

Theorem: Product of arguments

 * $\exp \left({ xy }\right) = \exp \left({ x }\right)^{y}$


 * 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:


 * $\dfrac {\mathrm d y} {\mathrm d x} = f \left({x, y}\right)$
 * $y \left({ 0 }\right) = 1$

on $\R$, where $f \left({x, y}\right) = y$.

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

 * $\exp \left({ x + y }\right) = \exp \left({ x }\right) \exp \left({ y }\right)$


 * Proof

Theorem: Product of arguments

 * $\exp \left({ xy }\right) = \exp \left({ x }\right)^{y}$


 * 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