feat(Analysis/Complex): residue theorem for rectangular contours

[jerwaynejones]

Adds Mathlib/Analysis/Complex/RectangleResidue.lean (644 LoC), filling the gap between Cauchy–Goursat for rectangles (Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable) and the Cauchy integral formula for circles. The file proves the residue formula for rectangular boundary integrals together with a rotated principal log branch needed to evaluate the left-edge integral that crosses the standard cut.

Why

Mathlib has Cauchy–Goursat for rectangles and the Cauchy integral formula for circles, but no residue theorem for rectangular contours. This file fills that gap. It was extracted from a downstream Lean 4 formalization project (a conditional proof of Goldbach's conjecture, where contour shifting on rectangles arises naturally in Perron's formula).

Main results

• Complex.hasDerivAt_log_neg — the rotated logarithm log_neg z := log (-z) + I * π has derivative z⁻¹ on the rotated slit plane {z | -z ∈ slitPlane} (cut on the positive real axis).
• Complex.boundaryIntegral_inv_sub_eq_two_pi_I — for ρ strictly inside the rectangle [z, w], ∮_{∂[z,w]} (s − ρ)⁻¹ ds = 2πi.
• Complex.boundaryIntegral_eq_residue_sum — parametric residue theorem for a sum of simple poles plus a holomorphic remainder, packaged as Complex.RectangleResidueData.
• Complex.boundaryIntegral_single_pole — the one-pole specialization.

Structure

The development proceeds in five stages:

1. A notation Complex.boundaryIntegral for the oriented boundary integral over [z, w], with a Cauchy–Goursat wrapper and additivity/scaling lemmas.
2. A rotated principal logarithm Complex.log_neg z := log (-z) + I * π with cut on the positive real axis, plus its derivative and corner-comparison identities with the standard Complex.log.
3. Explicit FTC-based evaluations of the four edge integrals of (s − ρ)⁻¹.
4. A closed-form residue boundary integral ∮_{∂[z,w]} (s − ρ)⁻¹ ds = 2πi for ρ strictly inside the rectangle.
5. A parametric residue theorem for sums of simple poles plus a holomorphic remainder, packaged as Complex.RectangleResidueData.

References

• L. Ahlfors, Complex Analysis (3rd ed.), §4.5.
• E. Stein and R. Shakarchi, Complex Analysis, Ch. 2 §3 Thm 3.1.

Status

Draft while CI verifies the build against current master — the file was developed against v4.28.0 and cherry-picked forward 2942 commits, so import surface changes are possible. Once CI is green I will flip to ready-for-review.

🤖 Generated with Claude Code

[github-actions]

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[github-actions]

PR summary 1b6d405437

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.Complex.RectangleResidue (new file)	2515

---

Declarations diff

+ RectangleResidueData
+ RectangleResidueData.toIntegrand
+ bottomEdge_integral_eq
+ boundaryIntegral
+ boundaryIntegral_add
+ boundaryIntegral_const_mul
+ boundaryIntegral_eq_residue_sum
+ boundaryIntegral_eq_zero_of_diffOn
+ boundaryIntegral_finset_sum
+ boundaryIntegral_inv_sub_assemble
+ boundaryIntegral_inv_sub_eq_two_pi_I
+ boundaryIntegral_single_pole
+ hasDerivAt_clog_neg_real
+ hasDerivAt_log_neg
+ horizEdge_integral_eq
+ intervalIntegrable_finset_sum
+ leftEdge_integral_eq
+ log_neg
+ log_neg_eq_log_add_two_pi_I_of_im_neg
+ log_neg_eq_log_of_im_pos
+ neg_left_edge_in_slitPlane
+ rightEdge_integral_eq
+ slitPlane_bottom_edge
+ slitPlane_right_edge
+ slitPlane_top_edge
+ topEdge_integral_eq

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to strong technical debt.

No changes to weak technical debt.