Complex Numbers

$z=a+ib$ can be split into $Rez=a$ and $IMz=b$.

Operations

• Addition follows the pattern$a+ib+c+id=(a+c)+i(b+d)$. Two complex numbers are equal $a+ib=c+id$ if their real and imaginary parts are equal ($a=c\text{and}b=d$).
• Multiplication of Reals with Complex Numbers: $(a+i0)\cdot (c+id)=ac+iad$
• Division: $\frac{c+id}{a+ib}=\frac{(c+id)(a-ib)}{(a+ib)(a-ib)}=\frac{ac+bd}{a^2+b^2}+i\frac{ad-bc}{a^2+b^2}$
• Conjugate: If $z=a+ib$ then $\bar{z}=a-ib$ is its conjugate
• Reciprocal: $\frac{1}{z}=\frac{1}{r}{e}^{-i\theta }$

Notes

Radial coordinates are used for complex analysis. $\theta =argz$ defines the argument of $z$. Connecting the conjugate and magnitude of the complex number: $z\cdot \bar{z}=|z{|}^2$

Euler’s Formula

Converts polar form back into cartesian: $e^{i\theta }=cos\theta +isin\theta$

Roots of Complex numbers

The collection of $n$ distinct complex numbers is

$$\sqrt{\text{[}}n]r{e}^{i\theta }=\sqrt{\text{[}}n]r\exp \text{[}i(\frac{\theta }{n}+k\frac{2\pi }{n})],k=0,1,2,...,n-1$$

in which each of them have the property to equal $z$ when raised to the $n$-th power. Such that all these complex numbers are the $n$-th roots of $z=r{e}^{i\theta }$. These are all distinct roots and there are $n$ many of them. The principal root where $-\pi <\theta <\pi$ is given by

$$\sqrt{\text{[}}n]rexp\text{[}i(\frac{\theta }{n})]$$

No Effect if $k=m+n$

In the case where $k=m+n$, both roots, ($k=m$ and $k=m+n$) are equal. This is because $n$ is a positive integer (counter $k$ depends on it) and just results in differing the root by $2\pi n$, which is the same as $2\pi (0,1,2,...)$

$$\sqrt{\text{[}}n]r\exp \text{[}i(\frac{\theta }{n}+\frac{2\pi (m+n)}{n})]=\sqrt{\text{[}}n]r\exp \text{[}i(\frac{\theta }{n}+\frac{2\pi m}{n})]$$

Roots in the Complex Plane

All nth roots of $r{e}^{i\theta }$ lie in a circle centered at the origin with radius equal to the real, positive, nth root of $r$. One of them has argument $w\_0=\theta /n$ the others are uniformly spaced around the circle with distance $\Delta \theta =2\pi /n$. The image shows the three cube roots of $z=r{e}^{i\theta }$.