User:Keith.U/Exposition of the Natural Exponential Function/Real
Jump to navigation
Jump to search
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.
Theorem: $\exp$ is continuous
- $\exp x$ as defined above is continuous.
Theorem: Sum of arguments
- $\map \exp {x + y} = \map \exp x \map \exp y$
Theorem: Product of arguments
- $\map \exp {x y} = \map \exp x^y$
Theorem: $\exp$ is strictly positive
- $\exp$ is strictly positive.
Theorem: $\exp$ is its own derivative
- $D \exp = \exp$
Theorem: $\exp$ is strictly increasing
- $\exp$ is strictly increasing
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.
Theorem: $\exp$ is continuous
- $\exp x$ as defined above is continuous.
Theorem: Sum of arguments
- $\map \exp {x + y} = \map \exp x \map \exp y$
Theorem: Product of arguments
- $\map \exp {x y} = \map \exp x^y$
Theorem: $\exp$ is strictly positive
- $\exp$ is strictly positive.
Theorem: $\exp$ is its own derivative
- $D \exp = \exp$
Theorem: $\exp$ is strictly increasing
- $\exp$ is strictly increasing
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.
Theorem: $\exp$ is continuous
- $\exp x$ as defined above is continuous.
Theorem: Sum of arguments
- $\map \exp {x + y} = \map \exp x \map \exp y$
Theorem: Product of arguments
- $\map \exp {x y} = \map \exp x^y$
Theorem: $\exp$ is strictly positive
- $\exp$ is strictly positive.
Theorem: $\exp$ is its own derivative
- $D \exp = \exp$
Theorem: $\exp$ is strictly increasing
- $\exp$ is strictly increasing
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.
Theorem: $\exp$ is continuous
- $\exp x$ as defined above is continuous.
Theorem: Sum of arguments
- $\map \exp {x + y} = \map \exp x \map \exp y$
Theorem: Product of arguments
- $\map \exp {x y} = \map \exp x^y$
Theorem: $\exp$ is strictly positive
- $\exp$ is strictly positive.
Theorem: $\exp$ is its own derivative
- $D \exp = \exp$.
Theorem: $\exp$ is strictly increasing
- $\exp$ is strictly increasing
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.
Theorem: $\exp$ is continuous
- $\exp x$ as defined above is continuous.
Theorem: $\exp$ is strictly positive
- $\exp$ is strictly positive.
Theorem: Sum of arguments
- $\map \exp {x + y} = \map \exp x \map \exp y$
Theorem: Product of arguments
- $\map \exp {x y} = \map \exp x^y$
Theorem: $\exp$ is strictly increasing
- $\exp$ is strictly increasing
Unified Approach
Theorem: Equivalence of definitions of $\exp$
- All definitions of $\exp$ hitherto are equivalent.