# Real exponential function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0.

mko-real-exponential-function · definition · v0.10.0



## 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

[Addition and strict positivity of the full exponential series](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-exponential-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Infinite series](https://ocme.evemisslab.com/en/mko/mko-infinite-series/) — Use the whole series definition.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Arguments are real.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — E is defined on all reals.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Partial sums have a limit obligation.
- [Radius of convergence of a power series](https://ocme.evemisslab.com/en/mko/mko-power-series-radius-of-convergence/) — The radius is infinity.
- [Factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) — Coefficients use n!.
- [Absolute convergence of series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) — Cauchy products and rearrangements need justification.
- [Binomial theorem](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/real-exponential-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-real-exponential-function/index.md)