Mathematics

Complex Analysis

Complex analysis studies functions of complex variables, revealing deep connections between analysis, geometry, and physics. Its techniques are essential for quantum field theory, string theory, statistical mechanics, and conformal field theory.

1. Complex Numbers and the Complex Plane

Complex Numbers

A complex number z ∈ ℂ has the form:

$$z = x + iy, \quad x, y \in \mathbb{R}, \quad i^2 = -1$$

- Real part: Re(z) = x
- Imaginary part: Im(z) = y
- Complex conjugate: $\overline{z} = x - iy$
- Modulus: $|z| = \sqrt{x^2 + y^2}$
- Argument: $\arg(z) = \theta$ where $z = |z|e^{i\theta}$

Polar Form

$$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$$

Euler's formula:

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Complex Plane

The complex plane (Argand diagram) identifies ℂ with ℝ² where z = x + iy corresponds to point (x, y).

- Addition: vector addition
- Multiplication: rotation + scaling ($r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$)

2. Holomorphic (Analytic) Functions

A function f: ℂ → ℂ is holomorphic (or analytic) at z₀ if it is complex differentiable in a neighborhood of z₀.

Complex Derivative

f is complex differentiable at z₀ if the limit exists:

$$f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$$

This limit must be independent of the direction of approach in the complex plane.

Cauchy-Riemann Equations

Write f(z) = u(x,y) + iv(x,y) where u, v are real-valued. f is holomorphic iff u and v satisfy:

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Examples of Holomorphic Functions

  • Polynomials: $p(z) = \sum_{n=0}^N a_n z^n$
  • Exponential: $e^z = e^x(\cos y + i\sin y)$
  • Trigonometric: sin z, cos z
  • Rational functions: p(z)/q(z) (holomorphic where q(z) ≠ 0)
  • Logarithm: log z (holomorphic on ℂ \ {z ≤ 0}, multi-valued)

Properties

Holomorphic functions are:

  • Infinitely differentiable (C^∞)
  • Equal to their Taylor series in their domain (analytic)
  • Determined by their values on any open set (identity theorem)
  • Conformal (angle-preserving) where f'(z) ≠ 0

3. Contour Integration and Cauchy's Theorem

Contour Integral

A contour γ is a piecewise smooth curve in ℂ. The contour integral is:

$$\int_\gamma f(z) dz = \int_a^b f(\gamma(t)) \gamma'(t) dt$$

Cauchy's Theorem

If f is holomorphic in a simply connected domain D and γ is a closed contour in D, then:

$$\oint_\gamma f(z) dz = 0$$

This is the fundamental theorem of complex analysis—holomorphic functions have path-independent integrals.

Cauchy's Integral Formula

If f is holomorphic inside and on a simple closed contour γ, then for any z₀ inside γ:

$$f(z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z - z_0} dz$$

For derivatives:

$$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z - z_0)^{n+1}} dz$$

Maximum Modulus Principle

If f is holomorphic and non-constant in a domain D, then |f| has no maximum in D. Maximum occurs only on the boundary.

4. Power Series and Laurent Series

Power Series

A power series centered at z₀ is:

$$f(z) = \sum_{n=0}^\infty a_n (z - z_0)^n$$

Converges inside a radius of convergence R. If f is holomorphic at z₀, it equals its Taylor series:

$$a_n = \frac{f^{(n)}(z_0)}{n!}$$

Laurent Series

For a function holomorphic in an annulus $r < |z - z_0| < R$, the Laurent series includes negative powers:

$$f(z) = \sum_{n=-\infty}^\infty a_n (z - z_0)^n$$

Coefficients:

$$a_n = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{(z - z_0)^{n+1}} dz$$

Singularities

If f is not holomorphic at z₀ but is holomorphic in a punctured neighborhood:

  • Removable singularity: a_n = 0 for n < 0 (can extend f to be holomorphic at z₀)
  • Pole of order m: a_{-m} ≠ 0 but a_n = 0 for n < -m
  • Essential singularity: infinitely many a_n ≠ 0 for n < 0

Example: $f(z) = \frac{1}{z^2}$ has a pole of order 2 at z = 0
$e^{1/z} = \sum_{n=0}^\infty \frac{1}{n! z^n}$ has an essential singularity at z = 0

5. Residue Theorem

The residue theorem is one of the most powerful tools in complex analysis and physics.

Residue

The residue of f at an isolated singularity z₀ is the coefficient a_-1 in the Laurent series:

$$\text{Res}(f, z_0) = a_{-1} = \frac{1}{2\pi i} \oint_\gamma f(z) dz$$

Residue Theorem

If f is holomorphic inside and on a simple closed contour γ except for isolated singularities z₁, ..., z_n inside γ:

$$\oint_\gamma f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$$

Computing Residues

For a simple pole at z₀:

$$\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$$

For a pole of order m:

$$\text{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}}\left[(z - z_0)^m f(z)\right]$$

Example: Real Integrals

Evaluate $\int_{-\infty}^\infty \frac{dx}{1 + x^2}$:

Consider $f(z) = \frac{1}{1 + z^2}$ with simple poles at z = ±i. Close the contour in upper half-plane:

$$\int_{-\infty}^\infty \frac{dx}{1 + x^2} = 2\pi i \cdot \text{Res}(f, i) = 2\pi i \cdot \frac{1}{2i} = \pi$$

6. Conformal Mappings

A conformal map is a holomorphic function that preserves angles (but not necessarily lengths).

Definition

f: D → ℂ is conformal at z₀ if f is holomorphic at z₀ and f'(z₀) ≠ 0. The map:

  • Preserves angles between curves
  • Scales lengths by |f'(z₀)|
  • Rotates by arg(f'(z₀))

Examples

Linear map: $f(z) = az + b$ (translation + rotation + scaling)

Möbius transformation:

$$f(z) = \frac{az + b}{cz + d}, \quad ad - bc \neq 0$$

Maps circles/lines to circles/lines. Examples:

  • $f(z) = 1/z$: inversion (swaps inside/outside of unit circle)
  • $f(z) = \frac{z - i}{z + i}$: maps upper half-plane to unit disk

Exponential map: $w = e^z$ maps horizontal strip to annulus/disk

Joukowski map: $w = z + 1/z$ (used in airfoil theory)

Riemann Mapping Theorem

Any simply connected domain D ⊂ ℂ (except ℂ itself) can be conformally mapped to the unit disk.

This is fundamental in 2D potential theory and conformal field theory.

7. Analytic Continuation and Riemann Surfaces

Analytic Continuation

If f is holomorphic on D₁ and g is holomorphic on D₂ with D₁ ∩ D₂ ≠ ∅, and f = g on D₁ ∩ D₂, then g is the analytic continuation of f.

By the identity theorem, analytic continuation is unique.

Multi-valued Functions

Functions like $\sqrt{z}$, log z, $z^{1/n}$ are multi-valued. To make them single-valued, introduce branch cuts.

Example: log z = ln|z| + i arg(z). Since arg(z) is multi-valued (arg(z) + 2πk), we choose a branch:

  • Principal branch: -π < arg(z) ≤ π, branch cut along negative real axis

Riemann Surfaces

A Riemann surface is a complex 1-dimensional manifold where multi-valued functions become single-valued.

Example: For $w = \sqrt{z}$, the Riemann surface has two sheets connected at z = 0 (branch point). Going around the origin once takes you from one sheet to the other.

Riemann surfaces are classified by genus g (number of handles):

  • g = 0: Riemann sphere (ℂ ∪ {∞})
  • g = 1: Torus (elliptic curves)
  • g ≥ 2: Higher genus surfaces

Relevance to Physics

- String theory worldsheets are Riemann surfaces
- Quantum field theory amplitudes involve integrals over moduli spaces of Riemann surfaces
- Elliptic curves (g=1) appear in integrable systems

8. Applications to Physics

Quantum Field Theory

Feynman propagators: Contour integration in complex energy plane, poles at ±E + iε

Wick rotation: Analytic continuation from Minkowski (t real) to Euclidean (τ = it) spacetime for path integrals

Dispersion relations: Causality + analyticity relates real and imaginary parts of scattering amplitudes

Conformal Field Theory

2D CFTs use complex coordinates z, z̄. Holomorphic and anti-holomorphic sectors separate:

$$T(z) = \text{holomorphic stress tensor}, \quad \overline{T}(\overline{z}) = \text{anti-holomorphic}$$

Conformal transformations are holomorphic maps. Correlation functions satisfy:

$$\langle \phi_1(z_1) \cdots \phi_n(z_n) \rangle$$

with specific transformation properties under conformal maps.

String Theory

Worldsheet: 2D surface parametrized by complex coordinate z
Vertex operators: V(z, z̄) create string states, holomorphic in z
Scattering amplitudes: Integrals over moduli space of Riemann surfaces

Virasoro algebra: $[L_m, L_n] = (m - n)L_{m+n} + \frac{c}{12}m(m^2 - 1)\delta_{m+n,0}$

Statistical Mechanics

Partition function: Analytic in complex temperature β = 1/kT
Lee-Yang theorem: Zeros of partition function lie on unit circle in complex fugacity plane
Critical phenomena: 2D Ising model solved using conformal field theory

General Relativity

Null coordinates: u = t - r, v = t + r become complex for analytic continuation
Penrose diagrams: Conformal compactification using Möbius transformations
Newman-Penrose formalism: Uses complex null tetrads and spin coefficients

Quantum Mechanics

WKB approximation: Analytic continuation to complex x
Tunneling: Contour integration under barriers in complex x-plane
Resurgence theory: Non-perturbative effects from Stokes lines

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