# MATH3401

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

$$*\, \Omega =$$ 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:

1. dilatation & rotation;
2. 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: $$\dfrac{1}{z}$$.