Improper integral

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

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 — Begin with finite-cutoff integrals.
• Function limits on punctured real domains — Take each cutoff to its boundary or infinity.
• Sequence limits and exact tail cutoffs — Discrete cutoff approximations have a limited scope.
• 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.