MATH529: L18 Functions of a complex variable

Overview

• $w=f\left(z\right)$ as a mapping between complex planes

• $w=f\left(z\right)$ as a flow

• Analysis in the complex plane:

  • limits

  • derivatives, differentiability

  • analytic functions

$w=f\left(z\right)$ mapping between complex planes

• $f:ℝ\to ℝ$, graph of $f$ as $\{(x,f\left(x\right)),x\in ℝ\}$, depicted by a plot

• $w:ℂ\to ℂ$, mapping from the $z$ complex plane to the $w$ complex plane

  • In the $z$-plane $z=x+iy$

  • In the $w$-plane $w=u+iv$, $w\left(z\right)=u(x,y)+iv(x,y)$

• Examples:

  • $w\left(z\right)=z^2$

  • $w\left(z\right)=z^4-3z$

  • $w\left(z\right)=(z^3-2)/(z^3+2)$

$w=f\left(z\right)$ flows in the complex plane

• $w\left(z\right)=u(x,y)+iv(x,y)$

• Interpret $(u,v)$ as components of a velocity vector in $ℂ$

• Streamlines are defined as solutions to the ODE system

  ${\displaystyle \frac{dx}{dt}=u(x\left(t\right),y\left(t\right)),\frac{dy}{dt}=v(x\left(t\right),y\left(t\right))}$

• Examples:

  • $w\left(z\right)=\bar{z}=x-iy$

    ${\displaystyle \frac{dx}{dt}=x,\frac{dy}{dt}=-y\Rightarrow x\left(t\right)=e^{t}x\left(0\right),y\left(t\right)=e^{-t}y\left(0\right)}$

$ℂ$-analysis: limits and continuity

• $f:ℝ\to ℝ$ $f\left(x\right)$ has limit $f_0$, if $\forall \epsilon >0$, $\exists {\delta }_{\epsilon }$ s.t. $|f\left(x\right)-f\left(x_0\right)|<\delta$ if $|x-x_0|<\epsilon$

  ${\displaystyle {lim}_{x\to x_0}f\left(x\right)=f_0}$

• $w:ℂ\to ℂ$ $w\left(z\right)$ has limit $w_0$ ($w$ might not defined at $z_0$), if $\forall \epsilon >0$, $\exists {\delta }_{\epsilon }$ .s.t $|w\left(z\right)-w\left(z_0\right)|<{\delta }_{\epsilon }$ if $|z-z_0|<\epsilon$.

  ${\displaystyle {lim}_{z\to z_0}w\left(z\right)=w_0}$

• If $w\left(z_0\right)=w_0={lim}_{z\to z_0}w\left(z\right)$, then $w\left(z\right)$ is continuous at $z_0$

$ℂ$-analysis derivatives and differentiability

• $f:ℝ\to ℝ$ is differentiable at $x_0$ if

  ${\displaystyle {lim}_{x\to x_0}\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}=f^{\text{'}}\left(x_0\right)}$

• Simple example: $f\left(x\right)=4+x$, $f^{\text{'}}\left(x_0\right)=1$

• $w:ℂ\to ℂ$ is differentiable at $z_0$ if

  ${\displaystyle {lim}_{z\to z_0}\frac{w\left(z\right)-w\left(z_0\right)}{z-z_0}=w^{\text{'}}\left(z_0\right)}$

• Simple example: $w\left(z\right)=x+3iy$, $z=x+iy$, $z_0=x_0+i{y}_0$

  ${\displaystyle \begin{array}{ll}\mathrm{along}x& \mathrm{along}y\\ {lim}_{x\to x_0}\frac{x+3i{y}_0-x_0-3i{y}_0}{x-x_0}=1\ne & {lim}_{y\to y_0}\frac{x_0+3iy-x_0-3i{y}_0}{y-y_0}=3i\end{array}}$

Analytic functions

• $w:ℂ\to ℂ$ is differentiable at $z_0$ if ${lim}_{z\to z_0}\frac{w\left(z\right)-w\left(z_0\right)}{z-z_0}=w^{\text{'}}\left(z_0\right)$

• $w:ℂ\to ℂ$ is analytic at a point $z_0$ if it is differentiable at all points in some neighborhood of $z_0$, $N_{\epsilon }\left(z_0\right)=\{z,|z-z_0|<\epsilon \}$

• $w:ℂ\to ℂ$ is analytic over a domain $D$ if it is analytical at all points within the domain

Cauchy-Riemann conditions

• $w:ℂ\to ℂ$, $w\left(z\right)=u+iv$, $z=x+iy$, $w$ is differentiable at $z_0$ iff

  ${\displaystyle \frac{\partial u}{\partial x}(x_0,y_0)=\frac{\partial v}{\partial y}(x_0,y_0),\frac{\partial u}{\partial y}(x_0,y_0)=-\frac{\partial v}{\partial x}(x_0,y_0)}$ ${\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x},\text{ or }{u}_x=v_y,u_y=-v_x}$

• Analogous to total differentials

  ${\displaystyle df(x,y)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy}$

• Given some differential form

  ${\displaystyle M(x,y)dx+N(x,y)dy}$

  it is a total differential if $M_y=N_x$