Complex numbers

Chapter 3 Complex functions
Chapter 3.1 Introduction
You know addition, subtraction, multiplication and division. But what about complex functions?
There are used in electricity, automatics,,,
-rational
-exponential
complex functions mainly
The other complex functions are used rarely and we will not work on them. The complex functions diagrams are more complicated than “normal”-real number functions diagrams. Every 2-d point is assigned with the complex 2-d function point. We step in 4-d space.
It’s difficult to imagine for everyman!

Chapter 3.2 Rational complex functions
This function is calcualated when addition, subtraction, multiplication and division are used only.

Fig. 3-1
Rational Complex Function

Chapter 3.3 Complex exponential function exp(z)
Chapter 3.3.1 Definition

Fig. 3-2
Complex exponential function

Chapter 3.3.2 Number e=2,7182818…
Irrational number e=2,71828.. is the fourth you should to know. The first are  0,1 and π. Everybody knows and origin*. π number is a “circumference/diameter ratio”. But number? Is any so simple defintion. There is , but not so obvious. Additionally, it came from the world of banks.
*0 and 1. Where are they from? It looks like an existential question.

Fig. 3-3
e=2,7182818…
number and bank interests
a. I established 1$ as a deposit in the very good bank. The inerests are 100% per year! Note that 1=100%
b. It should be 2$ after a 1 year
c. I agreed with a bank, that the interests is the same-100% per year, but I will receive the 1.5$ cash after half a year.
Capitalization is after after half a year in banker language.
d, The capitalization after after a 1 year but after capitalisations
e. The capitalization after a 1 year but after 3 capitalisations
f. The capitalization after a 1 year but after 10 capitalisations
g. The same as f. but in decimal formula
h. The capitalization after a 1 year but after 100 capitalisations
i. The capitalization after a 1 year but after 1 000 000 capitalisations
You can test the calculations by calculator
The more is more is capitalizations in a year –> we are near to the so called continuous capitalization
Conclusion:
e is a continuous capitalization :
-after one year
-when 100% interests
-when we invested 1$

Chapter  3.3.3 Rational exponential function exp(x)
There’s no problems with exp(x) calculations, when x=0,1,2…n

Fig. 3-4
exp(x)
exponencial function when x is a  natural number.
But what about any real number for example x=1.234?.
We do likewise all the scientific calculators. We exchange exp(x) into power series. Maclaurin series here:

Fig. 3-5
Exponential function exp(x) as a Maclaurin power series.
It’s possible to calculate any exp(x) when 4 fundamental mathematical operation are used only.

Chapter  3.3.4 Complex exponential function exp(z)
We exchange exp(z) into Maclaurin power series here:

Fig. 3-6
Exponential function exp(z) as a Maclaurin power series.
The complex number z is here, and not a real number as in the Fig. 3-5.

Chapter 3.3.5 exp(jωt) as a exp(z) special case
Fig. 3-6 formula is a exp(z) function definition. We can calculate any exp(z) with the aid of fundamental arithmetic operations. It isn’t difficult but very exhausting job! What is more! The diagram is hard to imagine too.–>see chapter. 3.1. Hence, we will study a special case when z=jωt for t=0…∞. This is exp(jωt) function ideal to analize the phenomenons from the electricity, automatics, acoustics and others.

Fig. 3-7
Exp(jt) exponential function i.e. exp(z) for z=jωt when ω=1/sec here, is so called pulsation.
The upper Im z green 0…∞ semi-axis is a exp(z) function domain. The red circle is a exp(jωt) independent function variable. The red circle points are calculated when green points j0, jπ/6,  jπ/2, j3π/2 i j2π are inserted into formula Fig. 3-6.
These red circle points are:
exp(j0)=+1
exp(jπ/6)=0.866..+j0.5
exp(jπ/2)=+j
exp()=-1
exp(j3π/2)=-j
exp(j)=+1
There are other intermediate and not labelled green points on the green 0…∞ semi-axis. The appropriate red circle points are calculated similarly. It means that red circle is a 0…j2π semi-axis exp(jt) function value. The point z=1+j0 did a 1 full rotation. The next rotations will be made for j2π…j4π, j4π…j6π… The complex  exp(jωt) is a periodical function!  Note that the real exp(x) function is a contradiction of the periodical function.

Chapter 3.3.6 How did red Fig. 3-7 circle arise?
Other words. How did we calculate exp(jt)?
We can’t use Euler formula
exp(jt)=cos(t)+jsin(t)
because we don’ know it yet.
Let’s try other possible methods.

exp(j0).
This is the easiest problem.
exp(j0)= exp(0)=+1 because  j0=0 is real number.

exp(jπ/6)
It isn’t so nice now! We have to use Fig. 3-6 formula with principal mathematical operations only.
This job is very exhausting so I propose WolframAlfa- the brilliant mathematical tool.
Call www.wolframalpha.com  and follow the picture instructions.
Note that program uses symbol instead of “electrical” j.
Write into dialog window first Maclaurin power series components.
The will calculate an approximation of the e(jπ/6).

*see chapter 2.3.2 in the article “Fourier Series”

Fig. 3-8
exp(jπ/6)
as first 6 Maclaurin power series components–>see Fig. 3-6
There were 6 Maclaurin power series components only, but the calculated value exp(jπ/6)≈0.866…+j0.5 is very near to the ideal value!
It’s better seen when polar coordinates are used. Compare
Calculated
r≈1.00003 
and θ≈29.9993º
Ideal (theoretical)
r=1 and θ=30º
Conclusion: exp(jπ/6) point lies on the circle r=1 and θ=30º.

The next red circle points will be calculated by the specialized exp(z) function.
This function:
1. has more compnents than 6
2
. uses the knowledge that is a periodical function

Let’s calculate!

exp(jπ/2)

Fig. 3-9
exp(jπ/2)
Ideal!

exp(jπ)

Fig. 3-10
exp(jπ)
This is reportedly the most beautiful mathematical formula!
exp(jπ)=-1 or another words exp(jπ)+1=0

exp(j3π/2)

Rys. 3-11
exp(j3π/4)

exp(j2π)

Rys. 3-12
exp(j2π)
Note, that exp(j2π)= exp(j0)=+1–>periodical function

Chapter 3.3.6 Euler formula
exp(jt)=cos(t)+jsin(t)
or more generally
exp(jωt)=cos(ωt)+jsin(ωt)
when ω jest is so called angular velocity or pulsation in radian/sec
It’s more known as angular version when α=ωt
exp(jα)=cos(α)+jsin(jα)

This is  XVIII century formula and has an easy geometrical interpretation

Fig. 3-13
z=exp(jα)=cos(α)+jsin(jα)
Test it for α=0, π/6, π/2, π,3π/2 and. The effect is the same as chapter 3.3.6.
This is complex number interpretation as a vector with length=|z| (module z) and an angular α. see chapter 2.3 multiplication
So any complex number is
z=|z|exp(jα)=|z|[cos(α)+jsin(jα)]
Note
The |z|=1 in the Fig. 3-13.

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