Agent Skills
› parcadei/Continuous-Claude-v3
› contour-integrals
contour-integrals
GitHub提供复变函数中围道积分的解题策略，包括积分类型选择、围道设置及留数定理计算。结合 Sympy 工具辅助求解极点与积分值，适用于数学分析与科学计算场景。
Trigger Scenarios
解决复变函数中的围道积分问题
需要计算实轴或单位圆上的定积分
使用留数定理求和
Install
npx skills add parcadei/Continuous-Claude-v3 --skill contour-integrals -g -y
SKILL.md
Frontmatter
{
"name": "contour-integrals",
"description": "Problem-solving strategies for contour integrals in complex analysis",
"allowed-tools": [
"Bash",
"Read"
]
}
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
- For integral_{-inf}^{inf} f(x)dx where f decays like 1/x^a, a > 1:
-
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.
Version History
- d07ff4b Current 2026-08-20 12:47
