Cis θ

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The function $\text{cis}(\theta)$ is a shorthand way of writing the equivalent expression $\cos(\theta)+i\sin(\theta)$ :

By definition: $\text{cis}(\theta)=\cos(\theta)+i\sin(\theta)$

This form simplifies complex arithmetic and allows for the study of complex analysis, as well as reduces the workload in writing the expressions.

Purpose

The use of trigonometric values to represent the real and imaginary portions of an associated complex number.

In the standard (rectangular) form, a complex number would be represented

$a+bi$

However, on a complex number plane, the 'a' (real value) is associated with the x-axis and the 'b' (imaginary value) is associated with the y-axis. Therefore, any complex number (represented as a coordinate pair on the plane) can be identified by its distance from the origin, r, and its vector, or angle, θ, above the positive x-axis.

Essentially, a coordinate $(a,b)$ which represents a complex number, is converted into a polar equivalent, $(r,\theta)$ .

In this way, all complex numbers can be written:

$a+bi=r\cdot\text{cis}(\theta)$

Where:

$a=r\cdot\cos(\theta)$

$b=r\cdot\sin(\theta)$

As long as the same $\theta$ and the same $r$ is used for each of the real and imaginary portions.

Therefore,

$r=\sqrt{a^2+b^2}$ , and

$\theta=\arctan\left(\tfrac{b}{a}\right)$

Arithmetic

Even though addition and subtraction of complex numbers in rectangular form is as easy as combining like terms, multiplication and division have always been a tedious process of determining reciprocals, if necessary, and distributing the product of two binomials

When complex numbers are written in polar form, on the other hand, addition and subtraction have always been a matter of converting the number back into rectangular, another tedious process.

The advantage of polar form, in terms of arithmetic operations, is that multiplication, division, and exponentiation are exceptionally simple.

De Moivre's Theorem

Any two complex numbers in polar form, $r_1\cdot\text{cis}(\theta_1)$ and $r_2\cdot\text{cis}(\theta_2)$ are multiplied as such:

$\big(r_1\cdot\text{cis}(\theta_1)\big)\cdot\big(r_2\cdot\text{cis}(\theta_2)\big)=(r_1\cdot r_2)\cdot\text{cis}(\theta_1+\theta_2)$

And divided:

$\frac{r_1\cdot\text{cis}(\theta_1)}{r_2\cdot\text{cis}(\theta_2)}=\frac{r_1}{r_2}\cdot\text{cis}(\theta_1-\theta_2)$

And exponentiated:

$\big(r\cdot\text{cis}(\theta)\big)^n=r^n\cdot\text{cis}(n\theta)$

One Value, Infinite Angles

The trigonometric functions of sine and cosine are cyclical (that is, periodic).

It is important to realize that any given complex number on a complex plane can be arrived at by rotating around the pole a multitude of times.

It is easier to see in polar form, the number $r\cdot\text{cis}(\theta)$ . The magnitude of the angle itself can be increased or decreased by complete rotations about the circle/pole to arrive at the same locale. This is a fundamental concept in trigonometry that extends into complex analysis.

A Note on Arithmetic

When performing arithmetic on polar complex numbers such that the angle of the solution is changed, it is essential to include the infinite number of rotations before manipulating the angle. In this way, multiple angles can be computed and thusly multiple complex solutions for the original arithmetic operation.