Unit 2955: Advanced Complex Analysis

1. Introduction to Complex Numbers

• Definition of complex numbers
• Algebraic operations: addition, subtraction, multiplication, division
• The complex conjugate and modulus
• The complex plane (Argand diagram)
• Polar form of complex numbers
  • Euler's formula
  • De Moivre's theorem

2. Complex Functions

• Functions of a complex variable: definitions and examples
• Limits and continuity in complex functions
• Complex differentiability
• Cauchy-Riemann equations
• Analytic (holomorphic) functions
  • Properties and examples

3. Contour Integration

• Introduction to contour integrals
• Parametrization of contours
• Cauchy's Integral Theorem
• Cauchy's Integral Formula
• Applications:
  • Evaluating real integrals using contour integration
  • Integral representations of functions

4. Residue Theory

• Singularities of complex functions
  • Classification: removable singularities, poles, essential singularities
• Residues:
  • Definition and calculation techniques
• Residue theorem
• Application of residue theorem to compute complex integrals
• Evaluation of definite real integrals via residues

5. Conformal Mappings

• Definition and properties of conformal mappings
• Examples of conformal maps (linear, inversion, exponential, logarithm)
• The Riemann Mapping Theorem
• Applications in solving boundary value problems
  • Mapping complicated domains to simpler ones

6. Power Series and Laurent Series

• Power series expansions of complex functions
• Radius and disk of convergence
• Differentiation and integration of power series
• Laurent series expansions
• Classification of singularities using Laurent series

7. Analytic Continuation

• Concept of analytic continuation
• Branch points and branch cuts
• Multi-valued functions and Riemann surfaces
• Techniques for extending domain of definition

8. Harmonic Functions

• Definition and properties
• Connection to analytic functions
• The Laplace equation in two dimensions
• Harmonic conjugates
• Maximum principle and mean value property

9. Special Functions in Complex Analysis

• Gamma function
  • Definition via integral, properties, functional equation
• Zeta function
  • Introduction and significance
• Bessel functions
  • Definition and applications
• Applications in physics and engineering contexts

10. Applications of Complex Analysis

• Physics: quantum mechanics, electromagnetism
• Engineering: signal processing, control theory
• Fluid dynamics: potential flow theory
• Other applications:
  • Solving differential equations
  • Evaluation of integrals

End of Unit Outline