# Complex Analysis > ----- - Skill: `neuralblitz/complex-analysis` (Agent Skill) - Install (CLI): `npx skillmds@latest add neuralblitz/complex-analysis` - Raw SKILL.md: https://api.skillmd.com/api/skills/neuralblitz/complex-analysis/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - Author: NeuralBlitz (https://skillmd.com/u/neuralblitz) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/neuralblitz/complex-analysis --- ----- ## name: complex-analysis description: > Expert complex analysis assistant for mathematicians and physicists. Use this skill whenever the user needs: help with complex functions, contour integration, residue theory, conformal mapping, or understanding analytic structure in the complex plane. Includes both pure theory and applications to physics and engineering. trigger: Any mathematical problem involving complex variables - from integration to potential theory to analytic number theory. license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics # Complex Analysis — Theory and Applications Covers: **Complex Numbers · Analytic Functions · Complex Integration · Residue Theory · Conformal Mapping · Series** ----- ## Complex Numbers ### Algebraic Form ``` z = x + iy ``` Where i² = -1, x = Re(z), y = Im(z). ### Polar Form ``` z = r(cos θ + i sin θ) = re^{iθ} ``` Where r = |z| = √(x² + y²), θ = arg(z). ### Complex Conjugate ``` z̅ = x - iy ``` Properties: - z z̅ = |z|² - Re(z) = (z + z̅)/2 - Im(z) = (z - z̅)/(2i) ### Argument ``` arg(z) = θ, z = |z|e^{iθ} ``` Multi-valued: arg(z) = θ + 2πk. ### Euler's Formula ``` e^{iθ} = cos θ + i sin θ ``` Useful identities: - e^{iπ} = -1 - e^{iπ/2} = i - cos θ = (e^{iθ} + e^{-iθ})/2 - sin θ = (e^{iθ} - e^{-iθ})/(2i) ### Roots of Unity Solutions to z^n = 1: ``` z_k = e^{2πik/n}, k = 0, 1, ..., n-1 ``` ### Complex Logarithm ``` log z = ln|z| + i arg(z) ``` Multi-valued: log z = Ln z + 2πik. Principal value: Log z = ln|z| + iArg(z), with -π < Arg(z) ≤ π. ### Complex Exponentiation ``` z^w = e^{w Log z} = e^{w(log|z| + i arg z)} ``` May be multi-valued. ----- ## Analytic Functions ### Complex Differentiability Function f is complex differentiable at z₀ if: ``` lim_{z→z₀} (f(z) - f(z₀))/(z - z₀) exists ``` This limit must be same from all directions. ### Holomorphic (Analytic) f is holomorphic on open set if complex differentiable at every point. Equivalent: complex analytic (power series expansion exists locally). ### Cauchy-Riemann Equations For f(z) = u(x,y) + iv(x,y) to be holomorphic: ``` ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x ``` Equivalently in complex form: ``` ∂f/∂z̅ = 0 (where ∂/∂z̅ = (1/2)(∂/∂x + i∂/∂y)) ``` ### Harmonic Functions If f = u + iv holomorphic, then u and v satisfy Laplace's equation: ``` ∇²u = 0, ∇²v = 0 ``` Conjugate harmonic functions. ### Entire Functions Holomorphic on entire complex plane. Examples: polynomials, e^z, sin z, cos z. ### meromorphic Functions Holomorphic except isolated poles. Examples: rational functions, tan z, cot z. ### Isolated Singularities | Type | Definition | Example | |------|------------|---------| | Removable | Limit exists | sin z / z at z=0 | | Pole | |z|→∞ → ∞ | 1/z at z=0 | | Essential | Neither | e^{1/z} at z=0 | **Casorati-Weierstrass**: Near essential singularity, f gets arbitrarily close to any complex value. ### Laurent Series Around isolated singularity: ``` f(z) = Σ_{n=-∞}^{∞} a_n (z - z₀)^n ``` - **Principal part**: terms with n < 0 - **Regular part**: terms with n ≥ 0 Classification: - Removable: principal part = 0 - Pole of order m: finite principal part - Essential: infinite principal part ----- ## Complex Integration ### Contour Integration ``` ∫_γ f(z) dz = ∫_a^b f(γ(t)) γ'(t) dt ``` ### ML-Estimate (Modulus Bound) ``` |∫_γ f(z) dz| ≤ M L ``` Where |f(z)| ≤ M on path of length L. ### Cauchy Integral Theorem If f is holomorphic on simply connected domain D: ``` ∮_C f(z) dz = 0 ``` For any closed contour C in D. ### Cauchy Integral Formula For f holomorphic inside and on C: ``` f^(n)(z₀) = (n!/(2πi)) ∮_C f(z)/(z - z₀)^{n+1} dz ``` For n = 0: ``` f(z₀) = (1/2πi) ∮_C f(z)/(z - z₀) dz ``` ### Deformation Invariance Homotopic contours give same integral. ### Winding Number For closed contour C about point a: ``` n(C,a) = (1/2πi) ∮_C dz/(z - a) ``` Integer = number of times C winds counterclockwise around a. ### Argument Principle ``` (1/2πi) ∮ f'(z)/f(z) dz = N - P ``` Where N = zeros, P = poles (counted with multiplicity). ----- ## Residue Theory ### Residue Definition Coefficient a_{-1} in Laurent expansion: ``` Res(f, z₀) = a_{-1} ``` ### Residue Formula for Pole of Order m ``` Res(f, z₀) = (1/(m-1)!) lim_{z→z₀} d^{m-1}/dz^{m-1}[(z - z₀)^m f(z)] ``` For simple pole (m = 1): ``` Res(f, z₀) = lim_{z→z₀} (z - z₀)f(z) ``` ### Residue Theorem For domain containing closed contour C: ``` ∮_C f(z) dz = 2πi Σ Res(f, z_k) ``` Sum over residues inside C. ### Using Residues for Integration 1. Identify singularities inside contour 2. Compute residues 3. Sum × 2πi ### Essential Integrals **Type 1**: ∫₀^{2π} R(sin θ, cos θ) dθ - Use z = e^{iθ}, dz = i z dθ **Type 2**: ∫_{-∞}^{∞} f(x) dx - Use upper/lower half-plane for real rational functions **Type 3**: ∫₀^{∞} x^α f(x) dx - Branch cuts needed ### Indented Contours For integrals with poles on real axis: - Small semicircles above/below - Take limit as radius → 0 ### Principal Value ``` PV ∮ f(z) dz = lim_{ε→0} (∮_{C_ε} f(z) dz) ``` Cauchy principal value exists even when simple pole on contour. ----- ## Series Expansions ### Taylor Series For holomorphic f at z₀: ``` f(z) = Σ_{n=0}^{∞} a_n (z - z₀)^n ``` Radius of convergence = distance to nearest singularity. ### Maclaurin Series Taylor series about 0. Common expansions: ``` e^z = Σ z^n/n! sin z = Σ (-1)^n z^{2n+1}/(2n+1)! cos z = Σ (-1)^n z^{2n}/(2n)! log(1+z) = Σ (-1)^n z^{n+1}/(n+1) (|z| < 1, z ≠ -1) (1+z)^α = Σ C(α,n) z^n (|z| < 1) ``` ### Radius of Convergence R = distance from center to nearest singularity. ### Region of Convergence | Series Type | ROC | |-------------|-----| | Taylor at interior point | Disk to nearest singularity | | Laurent (inner) | Annulus between singularities | ### Uniform Convergence On compact sets → can differentiate/integrate term-by-term. ### Weierstrass M-Test If |f_n(z)| ≤ M_n for all z in domain and ΣM_n converges, then Σf_n converges uniformly. ### Abel's Theorem If Σa_n converges, then: ``` lim_{r→1^-} Σa_n r^n = Σa_n ``` Power series converge at boundary except possibly at singular points. ----- ## Conformal Mapping ### Conformal (Biholomorphic) f is conformal if: - Holomorphic with non-zero derivative - Or anti-holomorphic with non-zero derivative (reflection) Angles preserved (except at critical points). ### Conformal Equivalence Two domains are conformally equivalent if there exists bijective holomorphic map between them. **Riemann Mapping Theorem**: Simply connected non-empty proper subdomain of C is conformally equivalent to unit disk. ### Möbius Transformations ``` f(z) = (az + b)/(cz + d), ad - bc ≠ 0 ``` Properties: - Map circles/lines to circles/lines - Cross-ratio preserved - 3 points determine transformation ### Mapping Examples | Map | Formula | Maps to | |-----|---------|---------| | Translation | z → z + a | Shift | | Rotation | z → e^{iθ}z | Rotate | | Scaling | z → az | Scale | | Inversion | z → 1/z | Reflect in unit circle | ### Schwarz-Christoffel Mapping Maps upper half-plane to polygon: ``` f(z) = A + C ∫ (z - z₁)^{-α₁} ... (z - z_n)^{-α_n} dz ``` Where α_i π = interior angle at vertex. ### Green's Function For domain D with boundary: ``` G(z, ζ) = log|z-ζ| - h(z,ζ) ``` Harmonic with boundary condition G = 0. ### Dirichlet Problem Find harmonic function with prescribed boundary values. Solved using conformal mapping to standard domain. ----- ## Entire and Meromorphic Functions ### Order of Entire Function ``` ρ = lim sup_{r→∞} log log M(r) / log r ``` Where M(r) = max|f(z)| on |z| = r. Examples: - Polynomial: ρ = 0 - e^z: ρ = 1 - e^{z²}: ρ = 2 ### Hadamard's Theorem For entire function of finite order ρ: ``` f(z) = e^{P(z)} Π(z) ``` Where P is polynomial of degree ≤ ρ, Π is product over zeros. ### Picard's Theorem Entire function misses at most one value (Picard). ### Mittag-Leffler Expansion For meromorphic function with poles {a_n}: ``` f(z) = g(z) + Σ [P_n(1/(z - a_n)) + c_n] ``` ### Partial Fractions For rational function: ``` f(z) = Σ A_n/(z - a_n) + polynomial ``` ### Weierstrass Product For zeros {a_n}: ``` P(z) = Π (1 - z/a_n) e^{z/a_n + ... + z^{m_n}/m_n a_n^{m_n}} ``` Ensures convergence. ----- ## Harmonic Functions ### Maximum Principle Non-constant harmonic function cannot have interior maxima/minima. ### Mean Value Property For harmonic u: ``` u(z₀) = (1/2π) ∫_0^{2π} u(z₀ + re^{iθ}) dθ ``` ### Poisson Integral For disk: ``` u(re^{iφ}) = (1/2π) ∫_0^{2π} u(e^{iθ}) P(r, φ-θ) dθ ``` Poisson kernel: P(r,θ) = (1 - r²)/(1 - 2r cos θ + r²) ### Schwarz Formula For holomorphic f = u + iv: ``` f(z) = (1/2πi) ∮ f(ζ) (ζ + z)/(ζ - z) dζ/ζ ``` ### Harnack's Inequality For positive harmonic functions: ``` (r-1)/(r+1) ≤ u(z)/u(0) ≤ (r+1)/(r-1) for |z| = r ``` ### Dirichlet Problem Solve Laplace's equation with boundary conditions. Solution: Poisson integral formula. ### Neumann Problem Solve with specified normal derivative on boundary. ----- ## Analytic Continuation ### Analytic Continuation Extend holomorphic function beyond original domain. **Schwarz reflection principle**: If u = 0 on real axis segment, extend by reflection. ### Indirect Continuation Via analytic path: if two domains overlap and functions agree on overlap, they give continuation. ### Monodromy Theorem If continuation along any path is independent of homotopy class, function extends to universal cover. ### Riemann Surfaces Function becomes single-valued on appropriate covering space. Example: log z → covers of punctured plane. ### Natural Boundary Domain where function cannot be analytically continued across any boundary point. Example: Σz^n has unit circle as natural boundary. ### Analytic Continuation of Power Series Original: f(z) = Σa_n(z - z₀)^n Extended via: different center gives different expansion (same function where overlap). ### Schwarz Reflection If f extends continuously to real interval and is real-valued there, can reflect: ``` f(z̅) = f(z)̅ ``` ----- ## Applications ### Evaluation of Real Integrals Use contour integration with keyhole, upper/lower half-plane. **Real rational**: Close in upper half-plane **Trigonometric**: z = e^{iθ} substitution **Half-plane**: Use Jordan's lemma ### Inverse Laplace Transform Use Bromwich integral: ``` f(t) = (1/2πi) ∫_{c-i∞}^{c+i∞} F(s) e^{st} ds ``` ### Fluid Dynamics Complex potential: w(z) = φ + iψ - φ = velocity potential - ψ = stream function Flow around cylinder with circulation. ### Electrostatics Complex potential for 2D electrostatic problems. Use conformal mapping to solve boundary value problems. ### Fourier Transform ``` F(ω) = ∫_{-∞}^{∞} f(t) e^{-iωt} dt ``` Use complex analysis for inversion, poles in complex plane. ### Number Theory Analytic continuation of ζ(s): ``` ζ(s) = Σ n^{-s} ``` Non-trivial zeros: s = ½ + it (critical strip). Functional equation relates ζ(s) to ζ(1-s). ----- ## Common Errors to Avoid - Forgetting that complex differentiability is more restrictive than real - Confusing isolated singularities and branch points - Incorrectly handling multi-valued functions (log, root) - Using Cauchy's theorem when function has singularities inside contour - Forgetting to include all residues - Confusing Taylor and Laurent series - Incorrect branch cut placement - Forgetting that conformal maps preserve angles, not orientations (anti-holomorphic) - Using Poisson formula without checking boundary regularity - Not considering that analytic continuation may not be unique (monodromy) ----- ## Key References - **Complex Analysis** by Ahlfors — Classic text - **Complex Analysis** by Stein & Shakarchi — Modern treatment - **Functions of One Complex Variable** by Conway — More detailed - **Visual Complex Analysis** by Needham — Geometric intuition