Complex Analysis
Complex Analysis Curriculum
This curriculum provides a structured overview of the study of complex analysis.
Foundations
Complex Numbers:
- The complex plane
- Polar form of complex numbers
- Powers and roots of complex numbers
- Stereographic projection
Analytic Functions:
- Functions of a complex variable
- Limits and continuity
- Derivatives and the Cauchy-Riemann equations
- Harmonic functions
Elementary Functions:
- Exponential function
- Logarithmic function
- Trigonometric functions
- Hyperbolic functions
Core Concepts
Complex Integration:
- Contour integrals
- Cauchy's Integral Theorem
- Cauchy's Integral Formula
- Morera's Theorem
Series Representation of Analytic Functions:
- Power series
- Taylor series
- Laurent series
- Singularities and zeros
Residue Theory:
- The Residue Theorem
- Calculation of residues
- Applications to evaluation of integrals
- Argument Principle
Further Topics
Conformal Mapping:
- Definition and properties
- Linear fractional transformations
- Schwarz-Christoffel transformation
- Applications to boundary value problems
Special Topics:
- Entire functions
- The Gamma function
- The Riemann Zeta function
- Analytic continuation
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