Course: Complex Analysis and Differential Equations

Complex Analysis

Complex numbers

  • Representation of complex numbers: coordinate form, polar form

  • Power and roots: De Moivre’s formula

    \begin{equation} (\mathrm{cos}\theta+i\mathrm{sin}\theta)^{n}=\mathrm{cos}\ n\theta+i\mathrm{sin}\ n\theta\nonumber \\ \end{equation}
Write each number in standard form
Compute the root.

Complex functions

Analytic functions

(a) Definitions

  • Complex functions: A complex function \(f\) is a function of the complex variable \(z=x+iy\) that results in a complex-valued output

    \begin{equation} f(z)=u(x,y)+iv(x,y)\nonumber \\ \end{equation}

    where \(u(x,y)\) and \(v(x,y)\) are real functions of two variables.

  • Continuous: A complex function is continuous at a point \(z_{0}\) if and only if for any neighborhood \(\mathcal{V}\) of \(f(z_{0})\), \(f^{-1}(\mathcal{V})\) is a neighborhood of \(z_{0}\).