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

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

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

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