Complex Analysis Lecture Notes

Introduction to Complex Variables

• Complex Variable Definition: A complex variable Z is defined as Z = x + iy, where x and y are real numbers.
• Complex Plane: The XY plane is referred to as the complex plane.
• Analytic Functions: Functions that are dependent only on the combination x + iy and not x - iy.
• Linearity: x and y are linearly independent.

Stereographic Projection

• Compactification of the Complex Plane: To address the infinite directions of the complex plane, we compactify it to the surface of a sphere (Riemann sphere).
• Definition: Stereographic projection maps points from the complex plane to the sphere.
  • The plane corresponds to the equator of the sphere.
  • The north pole represents the point at infinity.
• Coordinates on Sphere:
  • The coordinates on the sphere satisfy the constraint Z1^2 + Z2^2 + Z3^2 = 1.
  • Polar coordinates: Z1 = sin(θ)cos(φ), Z2 = sin(θ)sin(φ), Z3 = cos(θ).

Mapping Between Complex Plane and Sphere

• Mapping: Points in the complex plane correspond to points on the sphere through lines from the north pole.
  • Points inside the unit circle in the complex plane map to the southern hemisphere.
  • The origin maps to the south pole.
• Equations connecting coordinates:
  • y = 2Z2 / (1 - Z3)
  • x = 2Z1 / (1 - Z3)
• Inverse Mappings:
  • Z1 = 2x / (x^2 + y^2 + 1)
  • Z2 = 2y / (x^2 + y^2 + 1)
  • Z3 = (x^2 + y^2 + 1) / (x^2 + y^2 + 1)

Distance Definitions

• Great Circle Distance: The shortest distance on the sphere between two points.
• Cordal Distance: Defined for two points Z1 and Z2 in the complex plane:
  • D(Z1, Z2) = 2 * |Z2| / sqrt(|Z1|^2 + 1)(|Z2|^2 + 1)

Analytic Functions

• Definition: A function f(Z) = U(x, y) + iV(x, y) is analytic if it satisfies the Cauchy-Riemann conditions:
  • ∂U/∂x = ∂V/∂y
  • ∂U/∂y = -∂V/∂x
• Properties of Analytic Functions:
  • Must have first-order continuous partial derivatives.
  • The real and imaginary parts satisfy Laplace's equation.
• Harmonic Functions: The real and imaginary parts of analytic functions are harmonic functions.

Entire Functions

• Definition: A function that is analytic everywhere in the finite complex plane is called an entire function.
• Examples:
  • Polynomials are entire functions.
  • Exponential functions (e^Z) are entire.
• Non-Examples: Functions with singularities at infinity are not entire (ex: tan(Z)).
• Liouville's Theorem: If a function is analytic everywhere in the extended complex plane and has no singularities, it must be constant.

Conclusion and Further Notes

• Understanding limits and behaviors of analytic functions is critical.
• The study of singularities will be addressed in future lectures.