What is the role of Jordan's Lemma?

It ensures the arc integral over a large semicircle vanishes when the integrand decays to zero as |z| → ∞, allowing the contour integral to reduce to pole residues.

How do I decide which contour to use?

Match the integral type to the standard templates in the skill: semicircular contours for improper real integrals, unit-circle for circular trig integrals, and keyhole contours for branch cuts.

Can I automate pole and residue finding?

Yes. The skill provides Sympy_Poles and Sympy_Residue commands to locate poles and compute residues, which you can integrate into your workflow.

contour-integrals

npx machina-cli add skill parcadei/Continuous-Claude-v3/contour-integrals --openclaw

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SKILL.md

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Contour Integrals

When to Use

Use this skill when working on contour-integrals problems in complex analysis.

Decision Tree

Integral Type Selection
- For integral_{-inf}^{inf} f(x)dx where f decays like 1/x^a, a > 1:
  - Use semicircular contour (upper or lower half-plane)
- For integral involving e^{ix} or trigonometric functions:
  - Close in upper half-plane for e^{ix} (Jordan's lemma)
  - Close in lower half-plane for e^{-ix}
- For integral_0^{2pi} f(cos theta, sin theta)d theta:
  - Substitute z = e^{i theta}, use unit circle contour
- For integrand with branch cuts:
  - Use keyhole or dogbone contour around cuts
Contour Setup
- Identify singularities and their locations
- Choose contour that encloses desired singularities
- sympy_compute.py solve "f(z) = inf" to find poles
Jordan's Lemma
- For integral over semicircle of radius R:
- If |f(z)| -> 0 as |z| -> inf, semicircular contribution vanishes
Compute with Residue Theorem
- oint_C f(z)dz = 2pii * (sum of residues inside C)
- sympy_compute.py residue "f(z)" --var z --at z0

Tool Commands

Sympy_Residue

uv run python -m runtime.harness scripts/sympy_compute.py residue "1/(z**2 + 1)" --var z --at I

Sympy_Poles

uv run python -m runtime.harness scripts/sympy_compute.py solve "z**2 + 1" --var z

Sympy_Integrate

uv run python -m runtime.harness scripts/sympy_compute.py integrate "1/(x**2 + 1)" --var x --from "-oo" --to "oo"

Key Techniques

From indexed textbooks:

[Complex Analysis (Elias M. Stein, Ram... (Z-Library)] The keyhole contour and one small, connected by a narrow corridor. The interior of Γ, which we denote by Γint, is clearly that region enclosed by the curve, and can be given precise meaning with enough work. We x a point z0 in that If f is holomorphic in a neighborhood of Γ and its interior, interior.
[Complex Analysis (Elias M. Stein, Ram... (Z-Library)] For the proof, consider a multiple keyhole which has a loop avoiding In each one of the poles. Let the width of the corridors go to zero. Suppose that f is holomorphic in an open set containing a toy contour γ and its interior, except for poles at the points z1, .
[Complex Analysis (Elias M. Stein, Ram... (Z-Library)] CAUCHY’S THEOREM AND ITS APPLICATIONS The following denition is loosely stated, although its applications will be clear and unambiguous. We call a toy contour any closed curve where the notion of interior is obvious, and a construction similar to that in Theorem 2. Its positive orientation is that for which the interior is to the left as we travel along the toy contour.
[Complex Analysis (Elias M. Stein, Ram... (Z-Library)] Suppose that f is holomorphic in an open set containing a circle C and its interior, except for poles at the points z1, . The identity γ f (z) dz = 2πi N k=1 reszk f is referred to as the residue formula. Examples The calculus of residues provides a powerful technique to compute a wide range of integrals.
[Complex analysis an introduction to... (Z-Library)] Hint: Sketch the image of the imaginary axis and apply the argument principle to a large half disk. Evaluation of Definite Integrals. The calculus of residues pro¬ vides a very efficient tool for the evaluation of definite integrals.

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/complex-analysis/contour-integrals/SKILL.md

View on GitHub

Overview

This skill provides practical strategies for solving contour integrals in complex analysis, including how to choose contours, identify singularities, and apply standard tools. It covers semicircular, unit-circle, and keyhole contours, uses Jordan's lemma to drop arc contributions, and computes results with the residue theorem.

How This Skill Works

Follow a decision-tree workflow: select the integral type, set up a contour that encloses the desired poles, and verify Jordan's lemma to ignore arc terms. Then compute the integral as 2πi times the sum of residues inside the contour, using the included tooling to locate poles and residues.

When to Use It

When the integrand decays like 1/x^a with a>1, use a semicircular contour in the upper or lower half-plane to evaluate ∫_{-∞}^{∞} f(x) dx.
When the integrand contains e^{ix} or trig factors, close the contour in the upper half-plane for e^{ix} (Jordan's lemma) or in the lower half-plane for e^{-ix}.
When evaluating ∫_0^{2π} f(cos θ, sin θ) dθ, substitute z = e^{iθ} and use the unit circle contour.
When the integrand has branch cuts, employ a keyhole (or dogbone) contour around the cuts to capture the branch contributions.
When solving definite integrals via residues, pick a contour enclosing the needed singularities and apply the residue theorem to sum their contributions.

Quick Start

Step 1: Identify the integral type and choose the appropriate contour using the decision tree in the skill.
Step 2: Set up the contour so it encloses the desired singularities and justify the arc contributions (Jordan's lemma when applicable).
Step 3: Apply the residue theorem and use the provided tools to locate poles and compute residues (Sympy_Poles, Sympy_Residue) to obtain the value of the integral.

Best Practices

Identify all singularities and their locations to choose a contour that encloses the desired poles.
Use Jordan's lemma to justify that the semicircular arc contribution vanishes when f(z) decays to 0 as |z| → ∞.
Match the contour to the type of integral: semicircle for improper real integrals, unit circle for trig substitutions, keyhole for branch cuts.
Compute the integral with the residue theorem: integral equals 2πi times the sum of residues inside the contour.
Use the provided Sympy-based tools to find poles and residues and to verify results (Sympy_Poles, Sympy_Residue, Sympy_Integrate).

Example Use Cases

Evaluate ∫_{-∞}^{∞} dx/(x^2+1) using a semicircular contour enclosing i and -i.
Evaluate ∫_{-∞}^{∞} e^{ix}/(x^2+1) dx by closing the contour in the upper half-plane (Jordan's lemma).
Compute ∮_{|z|=1} z/(z^2+1) dz using residues at z = i and z = -i.
Evaluate ∫_{0}^{∞} dx/(x(1+x)) via a keyhole contour around the positive real axis.
Evaluate ∫_{0}^{2π} dθ /(a + b cos θ) by substituting z = e^{iθ} and using unit-circle contour.

Frequently Asked Questions

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