### 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:
*e*^{ic}=cos*c*+*i*sin*c* - De Moivre's formula:
(cos
*c*+*i*sin*c*)^{n}=cos(*nc*)+*i*sin(*nc*) - relation to vectors in
**R**^{2} - 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,
*n*th-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
- inner radius of convergence, outer radius of convergence
- 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

- mappings in
### Further topics

- Rouché's theorem
- Pickard's little theorem
- Pickard's great theorem

Last modified: November 28, 2022