Complex analysis

Complex analysis is a branch of mathematical analysis that examines functions of complex numbers. It has huge applications in other parts of mathematics and engineering.

Many topics from real analysis are extendible to the complex plane.

Studied in MAT290 — Advanced Engineering Mathematics.

Sub-topics

Foundations

• Complex number
  • De Moivre’s theorem
• Differentiability
• Analytic function
• Harmonic function

Contour integration

• Cauchy-Riemann equations
• Contour integral
  • Cauchy-Goursat theorem
  • Cauchy’s integral formula
  • ML inequality
• Jordan’s lemma

Sequences and series

• Power series
  • Taylor series
  • Laurent series
• Residue
  • Cauchy’s residue theorem
  • Great Picard theorem
  • Principal value integral
  • Indentation of contours

Resources

• Advanced Engineering Mathematics, by Dennis G. Zill, for an introductory treatment of the material
• Visual Complex Analysis, by Tristan Needham