Operations on complex numbers

Hello% username%!
I received quite a lot of reviews about the first part and tried to take them all into account.
In the first part I wrote about the addition, subtraction, multiplication and division of complex numbers.
If you don’t know it, run quickly to read the first part :-)
The article is framed in the form of a sharlagka, the history here is extremely small, mostly formulas.
Enjoy reading!

So, we turn to more interesting and slightly more complex operations.
I will tell you about the exponential form of the complex number,
exponentiation, square root, modulus, and pro sine and
cosine complex argument.
I think we should start with the module of a complex number.
A complex number can be represented on the coordinate axis.
Real numbers will be located along x, and imaginary along y.
This is called a complex plane. Any complex number, for example

$$

$$z = 6 + 8i$$

obviously can be represented as a radius vector:

The formula for calculating the module will look like this:

$$

$$r = | z | = \ sqrt (x ^ 2 + y ^ 2)$$

It turns out that the modulus of the complex number z will be equal to 10.
In the last part, I talked about two forms of recording a complex number:
algebraic and geometric. There is another illustrative form of recording:

$$

$$z = r \: e ^ {i \ phi}$$

Here r is the modulus of the complex number,
and φ is arctg (y / x) if x> 0
If x <0, y> 0 then

$$

$$φ = arctg (y / x) + \ pi$$

If x <0, y <0 then

$$

$$φ = arctg (y / x) - \ pi$$

There is a wonderful formula of Moivre, which allows you to build a complex number in
whole degree. It was discovered by the French mathematician Abrach de Muavre in 1707.
It looks like this:

$$

$$z ^ n = r ^ n {(cos (\ phi) + i * sin (\ phi))} ^ n$$

As a result, we can raise the number z to the power of a:

$$

$$z.x = | z | ^ a * cos (a * arctg (y / x))$$

$$

$$z.y = | z | ^ a * sin (a * arctan (y / x))$$

If your complex number is written in exponential form, then
You can use the formula:

$$

$$z ^ k = r ^ ke ^ {ik \ phi}$$

Now, knowing how the modulus of a complex number and the Moivre formula is, we can find
n root of complex number:

$$

$$\ sqrt [n] {z} = \ sqrt [n] {r} \; cos {\ frac {\ phi + 2 \ pi k} {n}} + i * sin {\ frac {\ phi + 2 \ pi k} {n}}$$

Here k is numbers from 0 to n-1
From this we can conclude that there are exactly n distinct roots of the nth
degrees from a complex number.
We turn to the sine and cosine.
Euler's famous formula will help us calculate them:

$$

$$e ^ {ix} = cos ({x}) + i * sin ({x})$$

By the way, there is still an Euler identity, which is private
case of the Euler formula for x = π:

$$

$$e ^ {iπ} + 1 = 0$$

We obtain the formulas for calculating the sine and cosine:

$$

$$sin \: z = \ frac {e ^ {ix} -e ^ {- ix}} {{2i}}$$

$$

$$cos \: z = \ frac {e ^ {ix} + e ^ {- ix}} {{2}}$$

At the end of the article we can not fail to mention the practical application of complex
numbers so that there is no question

surrendered these complex numbers?
Answer: in some areas of science without them in any way.
In physics, in quantum mechanics there is such a thing as a wave function, which itself is complex-valued.
In electrical engineering, complex numbers have found themselves as a convenient replacement for the diffs that inevitably arise when solving problems with linear AC circuits.
Zhukovsky's theorem (wing lift) also uses complex numbers.
And also in biology, medicine, economics, and many more.
I hope, now you know how to operate with complex numbers and you can
put them into practice.
If something in the article is unclear - write in the comments, I will answer.