# Aktosun, Spring 2023, Math 5322, Course Outline

• ### Complex Numbers

• algebra with complex numbers
• complex conjugation, real part, imaginary part
• reciprocal
• polar representation
• modulus, absolute value
• phase, argument, principal argument
• roots
• infinity
• Euler's formula:  eic=cosc+isinc
• De Moivre's formula:   (cosc+isinc)n=cos(nc)+isin(nc)
• relation to vectors in R2
• complex plane C, sets and regions in C
• neighborhood of a point, deleted neighborhood of a point
• interior point, exterior point, boundary point
• open set, closed set
• bounded set, unbounded set
• accumulation point of a set, limit point of a set
• closure of a set
• connected set, simply connected set
• region in the complex plane, domain in the complex plane

• ### Complex functions

• independent variable, dependent variable
• argument of a function, value of a function
• single-valued function, multi-valued function
• domain of a function, range of a function
• mappings in C
• limits, continuity
• differentiability, derivative
• analyticity
• representation as an infinite series
• existence of derivative
• D-bar derivative
• Cauchy-Riemann equations
• entire functions
• reflection principle for analytic functions
• singularities
• poles of finite order
• essential singularities
• harmonic functions, harmonic conjugates

• ### Elementary functions

• exponential function
• polynomial, rational function, square-root function, nth-root
• trigonometric functions
• hyperbolic functions
• logarithmic function
• inverse trigonometric and inverse hyperbolic functions
• exponential function with complex base
• logarithmic function with complex base
• single-valued branch of a (multi-valued) function
• principal branch

• ### Complex integrals

• curve, contour
• contour integrals
• antiderivative
• Cauchy integral theorem (Cauchy-Goursat theorem)
• Cauchy integral formula
• Morrera's theorem
• expressing the value of the derivative as a contour integral
• Liouville's theorem
• maximum modulus principle
• Jordan's lemma

• ### Infinite series

• sequences, convergence
• series, convergence
• absolute convergence, uniform convergence
• power series
• Laurent series
• analytic continuation, uniqueness of analytic continuation

• ### Residues and poles

• singularity
• pole of finite order
• essential singularity
• removable singularity
• residue
• residue theorem
• evaluation of improper integrals using the residue theorem
• evaluation of definite integrals involving sine and cosine with the help of the residue theorem
• argument principle

• ### Conformal mapping

• mappings in C
• conformal mapping
• preservation of angle
• applications of conformal mappings

• ### Further topics

• Rouché's theorem
• Pickard's little theorem
• Pickard's great theorem

Tuncay Aktosun aktosun@uta.edu