Complex Anaylsis

"Complex analysis is the only drug permitted by the nation."

Apart from the inherent difficulty of the study, if one does not find complex analysis interesting, they might need to reconsider their major in mathematics.

Extensions

Complex Space

• Definition of a Complex Number $z = x + iy$
  • Polar Representation of Complex Numbers $z = r e^{i \theta}$
  • Sign of Complex Numbers $\text{sign}$
  • Relationship Between the Powers of $i$ and the Exponential Function
  • Magnitude of a Complex Number Raised to an Imaginary Power is Always 1 $\left| r^{i \theta} \right| = 1$
• Conjugate of a Complex Number $\overline{z}$
• Topology in Complex Space
  • Non-openness of the Real Axis in the Complex Plane $\mathbb{R}^{\circ} \ne \mathbb{R} \subset \mathbb{C}$
• Riemann Sphere $\widetilde{\mathbb{C}}$

Complex Functions

• Complex Function $f : \mathbb{C} \to \mathbb{C}$
• Relationship Between Trigonometric and Exponential Functions
• Relationship Between Hyperbolic and Exponential Functions
• Multivalued Functions and Branches
• Generalized Binomial Coefficients for Complex Numbers
  • Negative Binomial Coefficients
• De Moivre’s Theorem
  • Derivation of Triple Angle Formulas Using De Moivre’s Theorem
• Eneström–Kakeya Theorem

Continuity and Derivatives

• Limits of Complex Functions
• Analytic Functions: Continuity and Differentiation of Complex Functions
  • Analytic Continuation
  • Wirtinger Derivatives
  • Inverse Function Theorem
• Cauchy-Riemann Equations
  • Conditions for the Inverse of the Cauchy-Riemann Equations to Hold

Singularities

• Types of Singularities in Complex Analysis
• Zeros in Complex Analysis
  • Zeros and Poles of Rational Functions

Integration

• Integration of Complex Functions
• ML Lemma
• Cauchy’s Theorem
• Shrinking Lemma for Complex Path Integrals
• Cauchy’s Integral Formula
  • Poisson Integral Formula
• Fresnel Integrals

Consequences of Cauchy’s Integral Formula

• Morera’s Theorem
• Liouville’s Theorem
• Gauss’s Mean Value Theorem
• Fundamental Theorem of Algebra
• Maximum Modulus Principle
• Schwarz Lemma
• Rouché’s Theorem

• Series

• Weierstrass M-test
• Derivation of Taylor Series Using Complex Analysis
• What is a Laurent Series?
  • Principal Part of Laurent Series and Classification of Singularities
  • Laurent Expansion of Cotangent and Cosecant
• Residue Theorem
• Residues at Poles
• Residues at Simple Poles
• Using the Residue Theorem to Calculate the Sum of Series for All Integers
  • Calculating the Sum of the Reciprocals of Squares Using Complex Analysis

Contour Integration

• Definite Integrals Using Trigonometric Substitution on the Complex Plane
• Improper Integrals of Rational Functions via Divergent Semi-Circular Complex Path Integrals
• Jordan’s Lemma
• Improper Integrals Using Jordan’s Lemma
• Improper Integrals Including Singularities on the Real Axis Using Jordan’s Lemma
• Improper Integrals of Multivalued Functions

Complex Geometry

Conformal Mapping

• What is Conformal Mapping?
  • Conformal Mappings Preserve Angles
• Bilinear Transformation
  • Circles are Invariant Under Bilinear Transformation in the Extended Complex Plane
  • Cross Ratio
• Inversion in Complex Analysis
• Conformal Mapping of a Semicircle to a Quadrant
• Mapping a Sector to a Circle with Conformal Mapping
• Mapping a Parabola to a Half-Plane with Conformal Mapping
• Exponential Function as a Conformal Mapping
• Trigonometric Functions as Conformal Mappings
• Joukowsky Transform
• Schwarz-Christoffel Transformation

References

• Osborne (1999). Complex variables and their applications