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

Preamble

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

Approach 1: Limit of a Series

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

Approach 2: Limit of a Sequence

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

Approach 3: Unique Continuous Extension

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

Approach 4: Inverse of $\ln$

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

Approach 5: ODE

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

Unified Approach

Theorem: Equivalence of definitions of $\exp$

All definitions of $\exp$ hitherto are equivalent.

Proof