MATH3401
Complex Analysis
Lecture 4
Roots of a complex number
For \(n>1\), \(n\)-th roots of \(z=re^{i\theta} \in \C_*\) means:
We want all \(w\in \C\) such that \(w^n=z\).
Notation: \(\exp(\xi)= e^{\xi}\).
Roots of a complex number
\(z\) has \(n\) distinct \(n\)-th roots:
\[{\small \begin{align*} & \left\{ r^{1/n}\exp\left(\frac{i\theta}{n}\right), r^{1/n}\exp\left(\frac{i\theta}{n} + \frac{i2\pi}{n}\right), \right. \\ & \quad \left. r^{1/n}\exp\left(\frac{i\theta}{n} + \frac{i4\pi}{n}\right), \ldots, r^{1/n}\exp\left(\frac{i\theta}{n} + \frac{i2(n-1)\pi}{n} \right) \right\} \end{align*}} \]Functions and mappings
\(\Omega \subseteq \C\): A function \(f:\Omega \ra \C\) can be viewed as a mapping on \(\Omega\), the domain of \(f\).
If \(\Omega \) is not specified, take \(\Omega \) as large as possible.
For example for \(f(z)=\dfrac{1}{z}\), the domain is \(\C_*\) and \(f:\C_*\ra \C\).
We can write: \(f:z \mapsto \dfrac{1}{z}\); or \(w=\dfrac{1}{z}\); or just \(\dfrac{1}{z}\).
Functions and mappings
Usual notation: \(w: (x,y) \mapsto (u,v)\) i. e., \[ \begin{align*} w(x+iy) &= u(x+iy)+iv(x+iy)\\ & \text{ or} & \\ w(x, y) &= u(x,y)+iv(x, y) \end{align*} \]
Functions and mappings
Functions and mappings
\(*\, \Omega = \) domain of \( f = \) dom\(\left(\,f\right)\).
\(*\,\) Range\(\left(\,f\right)= f\left(\Omega\right)\) \[ = \left\{ w: w=f(z)\text{ for some }z\in\Omega\right\}. \]
\(*\, f^{-1}\left(\xi \right) = \left\{ z\in\C: f(z)=\xi
\right\}\)
inverse of \(f\) (not \(1/f\)).
NOT
necessarily a function!
Functions and mappings
Example 1 - \(f(z)=\dfrac{1}{z}\).
dom\(\left(\,f\right) = \C_*\),
\( \;f^{-1}(\xi)=\dfrac{1}{\xi}\) is a function.
Functions and mappings
Example 2 - \(g(z)=\dfrac{1}{1-|z|^2}\).
dom\(\left(g\right) = \left\{ z:|z|\neq 1\right\}\),
\(g: \left\{ z:|z|\neq 1\right\}\ra \R\).
Inverse is not a function.
Check: If \(g(z)=\rho\), then \(g(e^{i\theta}z)=\rho\) for any \(\theta \in \R\). 📝
Functions and mappings
Example 3 - \(h(z)=z^2\).
\(h:\C\ra \C\),
inverse is not a function.
Aim: Get a geometric picture
of a given function
E. g. \(w = 1 + z\) moves each point one (unit) to the right.
E. g. \(re^{i\theta} \mapsto re^{i\left(\theta + \pi/2\right)}\) rotates through an angle of \(\pi/2\) in the positive (counterclockwise) direction about 0.
Basic idea with new mappings
Break it down into compositions of known/easy maps.
Example 1: Linear transformation
\(w=Az+b\) with \(A,b\in \C, A\neq 0\).
Split up into:
- dilatation & rotation;
- translation.
Basic idea with new mappings
Example 1 (cont): Linear transformation
(i) \(z\mapsto Az\): Write \(A=ae^{i\alpha}\) with \(\alpha\in \R, a \in \R_+\).
So \(re^{i\theta} \mapsto \left(a r\right)e^{i\left(\theta + \alpha \right)}\) geometrically dilates (expands or contracts) modulus by a factor of \(a=|A|\), and rotates through an angle of \(\alpha=\arg \,A\).
Basic idea with new mappings
Example 1 (cont): Linear transformation
(ii) \(z\mapsto z + b\): Write \(b=b_1+b_2 i\) with \(b_1,b_2\in \R\).
Then translate \(b_1\) to the right and \(b_2\) up.
\(z \mapsto A z + b\): compose (i) and (ii).
Note: domain is \(\C\).
Next: \(\dfrac{1}{z}\).