Complex Analysis

• Complex numbers.
• Algebraic form.
• Geometric representation of complex numbers.
• Polar form.
• De Moivre's formula.
• Roots of a complex number.
• Polar form.
• Roots of unity.
• Computation of roots in algebraic form.


• Equations of degree 2.
• Polynomial equations of higher degree.

• Functions of a complex variable.
  • Domains in .
  • Algebraic form for a function of a complex variable.
• Local properties of a function.
• Limits and continuity.
• Derivation.
• Cauchy-Riemann Equations.
• Harmonic functions.
• Analytic functions.
• A catalogue of analytic functions.
  • Exponential in basis .
  • Trigonometric functions.
  • Hyperbolic functions.
  • The logarithm of a complex number.
  • Analyticity of the logarithmic function.
  • Complex exponentials.
  • Inverse trigonometric functions.
  • Inverse hyperbolic functions.
• Integrals.
  • Line integral.
  • Theorem LM.
  • Cauchy's Theorems.
  • Cauchy's Integral Formula.
  • Generalization of Cauchy's Integral Formula.
  • Morera's Theorem.
• The theorems of Liouville and d'Alembert.
  • Liouville's Theorem.
  • The Fundamental Theorem of Algebra.
• Complex series.
  • Series of complex numbers.
    • Convergence and divergence.
    • Series with non negative real terms.
  • Series of functions of a complex variable.
    • Convergence and divergence.
  • Uniform convergence.
• Power Series.
  • Taylor Series.
  • Laurent Series.
  • Isolated singularities.
   
Residues and Integrals.
• Residues.
• Application to integrals.
• Integrals of the form .
• Conformal mappings.
  • Definition and characterization