# Improper integral

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

Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently.

mko-improper-integral · definition · v0.10.0



## Mathematical statement

A one-sided integral int_a^infinity f, from a to infinity, is the limit of finite-cutoff integrals as b→infinity; this slice declares f(x)=scale/x^p on [lower,infinity). The two-sided exact family declares f(x)=left_scale/|x|^p on x≤−1, f(x)=0 on −1<x<1 and f(x)=right_scale/x^p on x≥1; its tails must converge separately. An interior-singularity family explicitly gives both power-law pieces on [c−1,c+1] and splits at c. A symmetric principal value is insufficient. Absolute convergence is sufficient but not necessary, and apparently stable finite cutoffs do not prove convergence.





## Representative formula (exact TeX)

I(f,a,b)=L(F,b)





## Assumptions


- One-sided tails fix f(x)=scale/x^exponent on [lower,infinity); two-sided tails fix the zero-middle piecewise integrand stated above.
- The interior singularity is fixed at c, with f(x)=left_scale*(c−x)^(−alpha_L) on the left and f(x)=right_scale*(x−c)^(−alpha_R) on the right, on [c−1,c+1] and with 0≤alpha_L,alpha_R<1.
- Record every one-sided endpoint/singularity obligation separately; a principal value cannot impersonate an ordinary improper integral.





## Symbol scope


- I — improper-integral record
- f — integrand
- a — lower cutoff
- b — upper cutoff
- L — limit operator
- F — finite-cutoff integral





## Intuition, definition, and misconceptions

Every infinite endpoint or singularity creates its own one-sided convergence obligation.

Divergent sides may cancel symmetrically only in a principal-value sense.

Finite truncations do not quantify all later cutoffs.

Do not identify Cauchy principal value with the ordinary improper integral or infer convergence from a few close large-cutoff values.





## Adopted derivation guide

[Separate limits at every infinite endpoint and singularity](https://ocme.evemisslab.com/en/derivations/derivation-mko-improper-integral-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Signed polynomial definite integrals on finite intervals](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) — Begin with finite-cutoff integrals.
- [Function limits on punctured real domains](https://ocme.evemisslab.com/en/mko/mko-function-limit/) — Take each cutoff to its boundary or infinity.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Discrete cutoff approximations have a limited scope.
- [Absolute convergence of series](https://ocme.evemisslab.com/en/mko/mko-absolute-convergence-of-series/) — Absolute convergence and comparison provide background.





## Verification boundary

Exact computations cover the declared power laws only; general integrability and comparison need separate proofs.

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 Lebesgue improper integrals.





## 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/improper-integral.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-improper-integral/index.md)