Real exponential function

Mathematical statement

Terms at x are x^n/n!, while the power-series coefficients1/n! have ratio1/(n+1) tending to0, giving infinite radius and absolute convergence for every fixed real x. The addition law needs the binomial coefficient identity at every degree and absolute convergence for Cauchy-product rearrangement; one declared degree checks only one coefficient. E(0)=1 and E(x)E(−x)=1. The half inverse identity makes E(x/2) nonzero before E(x)=E(x/2)^2 proves strict positivity. A finite partial-sum computation is not the whole exponential function, even where a special value coincides. Derivatives, logarithms, ODE uniqueness, injectivity and exact rational values at nonzero rational inputs are not automatic claims of this node.

Representative formula (exact TeX)

E(x)=S(x)

Assumptions

• Partial sums use exact rational x and BigInt/rational factorial arithmetic.
• Addition-law certificates distinguish coefficient identities from the absolute-convergence rearrangement argument.

Symbol scope

• E — real exponential function
• x — real argument
• S — absolutely convergent power series

Intuition, definition, and misconceptions

Separate the whole series law from finite partial sums.

Factorial tails control global convergence.

Addition, inverse and positivity use whole-series properties rather than numerical approximations.

Do not promote a finite sum to a general exact exponential value or prove the addition law from a few x,y samples.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Infinite series — Use the whole series definition.
• Real number — Arguments are real.
• Function mapping — E is defined on all reals.
• Sequence limits and exact tail cutoffs — Partial sums have a limit obligation.
• Radius of convergence of a power series — The radius is infinity.
• Factorial — Coefficients use n!.
• Absolute convergence of series — Cauchy products and rearrangements need justification.
• Binomial theorem — Combine addition-law coefficients.

Verification boundary

This slice does not materialize logarithms, differential equations, derivative uniqueness or transcendence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add the standard normal distribution and differentiability of the real exponential.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.