Complex Analysis

Lecture 1: Algebra of complex numbers.

• Integration and differentiation of functions of complex variables
• Geometric interpretation of a complex number
• Trigonometric representation of a complex number
• Exponential representation of a complex number
• Practice with an exponential representation of a complex number
• Differentiation of functions of complex variables. Cauchy-Riemann conditions
• Practice with Cauchy-Riemann conditions
• Introduction to conformal mappings. Integration

Lecture 2: Cauchy theorem. Types of singularities. Laurent and Taylor series.

• Cauchy integral theorem
• Cauchy integral formula
• Taylor series in the complex plane
• Laurent series
• Types of singularities

Lecture 3: Residue theory with applications to computation of complex integrals.

• Integration with residues I
• Residue at infinity
• Jordan's lemma
• Integration with Jordan's lemma
• Integration in principal value

Lecture 4: Multivalued functions and regular branches.

• Extraction of the regular branch of the power type function
• Extraction of the regular branch of the log function
• Practice with regular branches

Lecture 5: Analytical continuation and Riemann surfaces.

• More on analytical continuation. Simple example
• Formal definition and uniqueness of analytic continuation
• Practice with analytic continuation: contour deformation
• Riemann surfaces [theory]
• Riemann surfaces [example]

Lecture 6: Integrals containing multivalued functions.

• Integrals with power-type integrand and two branch points.
• Integrals with log-type function
• The second type of integrals with the log-function
• Integrals with asymmetric integrand and log function