Alphabeta Math

The Exponential Function

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✓ 17 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

The real exponential function and the number e by a power series

Lemma

The exponential series converges absolutely for every real argument

Theorem

The exponential addition formula exp(x+y)=exp(x)exp(y)

Corollary

The exponential is positive and satisfies exp(-x)=1/exp(x)

Theorem

The exponential function is smooth and (exp)'=exp

Theorem

The exponential function is strictly increasing

Theorem

The exponential tends to +∞ at +∞ and to 0 at -∞

Corollary

The exponential is a continuous bijection from ℝ onto (0,∞)

Theorem

The exponential dominates every fixed nonnegative integer power at +∞

Theorem

The exponential is the unique solution of y'=y with y(0)=1

Theorem

Regular normalized multiplicative Cauchy equations characterize the exponential

Lemma

For fixed k, C(n, k)/nᵏ tends to 1/k!

Theorem

For every real x, (1+x/n)ⁿ→exp x

Theorem

Picard iteration from 1 produces the exponential partial sums

Theorem

The power-series, product-limit, IVP, functional-equation, and Picard definitions agree

Lemma

A geometric bound for tails of the exponential series

Corollary

The elementary numerical bound 2<e<3

Theorem

The number e is irrational

Remark

The exponential roadmap and its circularity hazards

Sources