Operations on complex numbers

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    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 this, you’d rather run to read the first part :-) The
    article is framed in the form of a spike, there are very few stories here, mostly formulas.
    Enjoy reading!

    So, we turn to more interesting and slightly more complex operations.
    I will talk about the exponential form of the complex number,
    exponentiation, square root, module, and also about the sine and
    cosine of the 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, you can think of it as a radius vector: The

    formula for calculating a module will look like this:

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


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

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


    Here r is a module of a 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 to an
    integer power. 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:

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


    $ zy = | 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 Moivr's formula are found, we can find the
    n root of a 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 different roots of the n-th
    degree 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 still exists the Euler identity, which is a special
    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, one cannot but mention the practical application of complex
    numbers, so that the question does not arise, did
    image
    these complex numbers surrender?
    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 can
    put them into practice.
    If something in the article is not clear - write in the comments, I will answer.
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