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analytic-functions

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Analytic Functions

When to Use

Use this skill when working on analytic-functions problems in complex analysis.

Decision Tree

  1. Is f analytic at z0?

    • Check Cauchy-Riemann equations: du/dx = dv/dy, du/dy = -dv/dx
    • Check if f has power series expansion around z0
    • Check if f is differentiable in neighborhood of z0
    • sympy_compute.py diff "u" --var x and sympy_compute.py diff "v" --var y

  2. Cauchy-Riemann Verification

    • Write f(z) = u(x,y) + iv(x,y)
    • Compute partial derivatives
    • Verify: du/dx = dv/dy AND du/dy = -dv/dx
    • z3_solve.py prove "cauchy_riemann"

  3. Power Series

    • f(z) = sum_{n=0}^{inf} a_n (z - z0)^n
    • Radius of convergence: R = 1/limsup |a_n|^(1/n)
    • sympy_compute.py series "f(z)" --var z --at z0

  4. Analytic Continuation

    • Extend f beyond original domain via power series
    • Identity theorem: if f = g on set with limit point, then f = g everywhere

Tool Commands

Sympy_Diff_U

uv run python -m runtime.harness scripts/sympy_compute.py diff "u(x,y)" --var x

Sympy_Diff_V

uv run python -m runtime.harness scripts/sympy_compute.py diff "v(x,y)" --var y

Sympy_Series

uv run python -m runtime.harness scripts/sympy_compute.py series "exp(z)" --var z --at 0

Z3_Cauchy_Riemann

uv run python -m runtime.harness scripts/z3_solve.py prove "diff(u,x) == diff(v,y)"

Key Techniques

From indexed textbooks:

  • [Complex Analysis (Elias M. Stein, Ram... (Z-Library)] A deep theorem which we prove in the next chapter says that the converse is true: every holomorphic function is analytic. For that reason, we use the terms holomorphic and analytic interchangeably. PRELIMINARIES TO COMPLEX ANALYSIS Corollary 2.
  • [Complex Analysis (Elias M. Stein, Ram... (Z-Library)] Cauchy, 1826 There is a general principle in the theory, already implicit in Riemann’s work, which states that analytic functions are in an essential way charac- terized by their singularities. That is to say, globally analytic functions are “eectively” determined by their zeros, and meromorphic functions by their zeros and poles. While these assertions cannot be formulated as precise general theorems, there are nevertheless signicant instances where this principle applies.
  • [Complex analysis an introduction to... (Z-Library)] EXERCISES If f(z) is analytic in the whole plane and real on the real axis, purely imaginary on the imaginary axis, show that f{z) is odd. COMPLEX INTEGRATION In the same situation, if v is the imaginary part of an analytic function f(z) in 12+, then f(z) has an analytic extension which satisfies f(z) = f(z). For the proof we construct the function V(z) which is equal to v(z) respect to this disk formed with the boundary values V.
  • [Complex analysis an introduction to... (Z-Library)] E is compact it can be covered by a finite number of the smaller disks, and we find that the p(/nJ are bounded on E, contrary to assumption. EXERCISES Prove that in any region 0 the family of analytic functions with positive real part is normal. Under what added condition is it locally bounded?
  • [Complex Analysis (Elias M. Stein, Ram... (Z-Library)] Notice that the radius of convergence of the above series is 1. Show that f cannot be continued analytically past the unit disc. Hint: Suppose θ = 2πp/2k, where p and k are positive integers.

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/analytic-functions/SKILL.mdView on GitHub

Overview

This skill provides practical strategies for solving problems involving analytic functions in complex analysis. It covers checking analyticity with Cauchy-Riemann, using power series, and applying analytic continuation to extend functions beyond their original domain. Stepwise methods and tool commands are included to support rigorous verification.

How This Skill Works

Technically, you verify analyticity by expressing f as u plus i v, assessing CR equations, and testing differentiability in a neighborhood. You then use power series expansions and radius of convergence to characterize the domain, followed by analytic continuation guided by the identity theorem.

When to Use It

  • When you need to verify that f is analytic at a specific point z0
  • When you want to confirm Cauchy-Riemann conditions hold for f
  • When you must compute or confirm a power series expansion around z0
  • When extending a function beyond its original domain via analytic continuation
  • When determining the radius of convergence and planning series computations

Quick Start

  1. Step 1: Represent f as u(x,y) + i v(x,y) and test Cauchy-Riemann
  2. Step 2: Compute partial derivatives and verify du/dx equals dv/dy and du/dy equals -dv/dx
  3. Step 3: If CR holds, compute the power series around z0 and inspect the radius of convergence

Best Practices

  • Write f in terms of u(x,y) and v(x,y) and check Cauchy-Riemann equations
  • Use symbolic tools to calculate partial derivatives and verify CR conditions
  • Compute a power series around z0 and determine radius of convergence
  • Apply the identity theorem to justify analytic continuation from a set with a limit point
  • Utilize available tool commands to automate differentiation and series computations

Example Use Cases

  • Checking if a candidate complex function is analytic at a domain point in coursework
  • Deriving the Taylor series for a complex function and identifying its radius of convergence
  • Extending an analytic function beyond its initial region using analytic continuation
  • Verifying CR conditions with symbolic differentiation in a problem set
  • Justifying continuation using the identity theorem when two definitions agree on a limit point

Frequently Asked Questions

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