Course: Complex Analysis and Differential Equations

Complex numbers

• Representation of complex numbers: coordinate form, polar form

• Power and roots: De Moivre’s formula

$$(\mathrm{cos}\theta+i\mathrm{sin}\theta)^{n}=\mathrm{cos}\ n\theta+i\mathrm{sin}\ n\theta\nonumber \\$$
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

$$f(z)=u(x,y)+iv(x,y)\nonumber \\$$

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 $$z_{0}$$.