Course: Complex Analysis and Differential Equations

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Representation of complex numbers: coordinate form, polar form

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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}

(a) Definitions

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Complex functions: A complex function \(f\) is a function of the

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

\(z=x+iy\) that results in a complex-valued outputwhere \(u(x,y)\) 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 \(z_{0}\).